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Nonparametric estimation of a survival function under progressive Type-I multistage censoring
Journal of Statistical Planning and Inference 141 (2011) 910–923
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Nonparametric estimation of a survival function under progressive Type-I multistage censoring$ M.D. Burke Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada T2N 1N4
a r t i c l e i n f o
abstract
Article history: Received 19 May 2009 Received in revised form 12 August 2010 Accepted 19 August 2010 Available online 24 August 2010
Failure time data subject to three progressive Type-I multistage censoring schemes are studied. Product limit estimators are proposed for the estimation of the survival function. It is shown that the resulting estimators are asymptotically equivalent to the corresponding estimators where the data are subject to a random iid right censoring scheme. Many well-known results on confidence bands and tests for randomly right censored data hold for these progressive censoring schemes. & 2010 Elsevier B.V. All rights reserved.
Keywords: Progressive multistage censoring Martingales Product limit estimation Asymptotic distribution
1. Introduction Experimental models using progressive censoring can be used in reliability analysis, product testing and clinical trials. In many situations, the precise estimation of the left tail of the distribution of failure time is of great importance. Hence, one would wish, at the beginning of the study, to test a large number of items (patients). In the reliability setting, to reduce the total time on test, the experimenter, at specified times, purposely and randomly removes from the test items that are still functioning. This is called progressive Type-I multistage censoring and it is the subject of this paper. One reason to purposely remove items is that they are costly to produce. A large proportion of them can be removed at a predetermined time. The items so removed can then be sold and re-used. Another reason to purposely remove functioning items is that the operation of the test may be costly. A third reason is that one may need to free up some laboratory space for other purposes. The removal (censoring) times can be chosen to correspond to important times such as various warranty periods. At these and other removal times, the removed items can be disassembled and examined for wear using destructive testing methods. We refer to Balakrishnan and Aggarwala (2000) for further applications in reliability. In clinical trials, restricting patient loads for physicians may be a factor in keeping costs down. Also, physicians may desire to spend more resources on new patients than on patients who have survived past certain benchmarks. Thus, removing all but a certain number of patients from the study at certain times is desirable. In this paper, we will develop an asymptotic theory for three types of progressive censored data. It is shown that martingale theory can be applied to these progressively censored data (Section 3). In addition, it is shown that the martingale processes for such data are asymptotically equivalent to a martingale process for data which are randomly right censored by independent and identically distributed random variables with a particular distribution function (Theorem 1).
$
Research partially supported by a Canadian NSERC Discovery Grant.
Tel.: +1 403 220 3959.
E-mail address: [email protected] 0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.08.009
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
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Such randomly right censored data have been studied extensively (Fleming and Harrington, 1991; Andersen et al., 1993). Using this asymptotic equivalence, many well-known results on confidence bands and tests for randomly right censored data are derived for these progressive censoring schemes. The results are applied to the nonparametric estimation of the survival function under progressive censorship and, in particular, the product limit estimator. Suppose there are n items that are put in use at time t ¼ 0. For item i, let T0i be the time to failure. We will consider three progressive censoring schemes and a corresponding random right censoring scheme. Let t1 o t2 o otm be fixed times when some items will be removed from study (censored). Assume that the T0i are independent random variables with a common continuous distribution function F0, cumulative hazard function L0 , and survival function S0 ¼ 1F0 . Let tF0 ¼ infft : F0 ðtÞ ¼ 1g. We will also assume that the T0i are, at first, subject to random right censorship by an independent sequence C0i of iid random variables with distribution function G0. If no progressive censoring occurs, one observes Ti ¼ Ti0 4C0i and d0i ¼ I½Ti0 r C0i . The survival function of the Ti is SðtÞ ¼ P½Ti 4t ¼ ð1G0 ðtÞÞS0 ðtÞ:
ð1Þ
The Ti have support ½0, tF , where tF ¼ infft : FðtÞ ¼ 1g, F ¼ 1S. Let bxc denote the greatest integer less than or equal to x and dxe denote the smallest integer greater than or equal to x. The ðTi , d0i Þ are subject to progressive censoring. The three such progressive schemes are: Scheme 1: (Progressive Type-p). For the n‘ items still observed to be functioning at time t‘ , randomly select, independent of the future values of Ti, bn‘ p‘ c of them to be removed (censored), where p‘ is a predetermined proportion, 0 r p‘ o1. Scheme 2: (Progressive Type-R). For the n‘ items still observed to be functioning at time t‘ , randomly select, independent of the future values of Ti, R‘ of them to be removed (censored). If fewer than R‘ items are on trial at t‘ , remove all the remaining ones. Scheme 3: (Progressive Type-R*). For the n‘ items still observed to be functioning at time t‘ , randomly select, independent of the future values of Ti, n‘ ðR‘ 4n‘ Þ of them to be removed (censored). (If n‘ rR‘ , no items are removed.) In Schemes 1 and 2 the proportion p‘ or the number R‘ of items to be removed are specified. In Scheme 3, one specifies the number R‘ to remain in the study at time t‘ . This is useful when one wants to ensure that there are sufficient items on trial for the asymptotics to hold. For Schemes 2 and 3, R‘ and R‘ are predetermined fixed numbers. If item i is removed at time t‘ under Scheme j, j ¼ 1,2,3, then let Cji ¼ t‘ . If Ti is observed, then we will put Cji ¼ tF , so that events of the form ½t o Cji ¼ ½Cji % / t, ½t rCji ¼ ½Cji 5t and ½Ti r Cji ¼ ½Cji 5Ti are well defined. One observes Xji ¼ minfTi , Cji g, dji ¼ I½Xji ¼ Ti and dji d0i ¼ I½Xji ¼ Ti0 , i ¼ 1,2, . . . ,n. The schemes above are Type-I censoring. However, they are different from the Type-I scheme in Bagdonavicˇius and Nikulin (2002, p. 79) where the censoring variables are predetermined non-random values and a special case of right censoring by independent random variables. The book by Balakrishnan and Aggarwala (2000) studies order statistics and parametric inference for progressive Type-II multistage censoring, where the times of censoring are order statistics of the lifetimes and the numbers removed are specified in advance. As pointed out by them (see p. 10), Type-I, and not Type-II censoring, is usually carried out in practice. Bordes (2004) established large sample properties of the product limit estimator under progressive Type-II censoring. The nature of the processes and their limit distributions are different in the Type-I and Type-II cases. In order to develop a proper asymptotic result for the Type-R and Type-R* models, we will assume that the numbers R‘ and R‘ are proportional to n, the sample size. (If R‘ is fixed and n tends to infinity, the effect of censoring is asymptotically negligible.) Note that if too many items are removed, the remaining items may be too few in number for the asymptotic theory to hold after time t‘ . In this case, inference about F0 would be valid up to time t‘ 4tF0 , only. We will assume that the R‘ of Scheme 2 satisfy R‘ ¼ bnr ‘uc,
‘ ¼ 1,2, . . . ,m,
ð2Þ
where the r‘u Z 0 are predetermined. For Scheme 3, we will assume R‘ ¼ dnr ‘ e, ‘ ¼ 1,2, . . . ,m
and
0 o r‘ or‘1 o1, ‘ ¼ 2, . . . ,m:
ð3Þ
Note that if r‘1 r r‘ , then no censoring will take place at t‘ . Let G denote the discrete (sub)-distribution function with probability mass function
dGðt1 Þ ¼ p1 ,
dGðt‘ Þ ¼ p‘
‘1 Y iu ¼ 1
ð1piu Þ,
dGðtF Þ ¼ 1
m X
dGðt‘ Þ,
ð4Þ
‘¼1
‘ ¼ 2, . . . ,m. If tF ¼ 1 and dGðtF Þ 4 0, then G is a sub-distribution function. Define the function p by
pðtÞ ¼ SðtÞð1GðtÞÞ, t Z0,
ð5Þ
where S is defined by (1). One can define an iid random censoring scheme. Scheme 4: (IID random). Let C4i be a sequence of independent and identically distributed random variables having distribution function G. One observes X4i ¼ minfTi ,C4i g, d4i ¼ I½X4i ¼ Ti and d4i d0i ¼ I½X4i ¼ Ti0 , i ¼ 1,2, . . . ,n.
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Using counting process notation, for j ¼ 1,2,3,4 and i ¼ 1,2, . . . ,n, let N0ji ðtÞ ¼ I½Ti0 r t,Ti0 r C0i 4Cji ;
N 0j ¼
n X
N0ji ;
i¼1
Y0ji ðtÞ ¼ I½t r Ti0 4C0i 4Cji ;
n X
Y 0j ¼
Y0ji ;
i¼1
A0ji ðtÞ ¼
Z 0
t
Y0ji ðuÞ dL0 ðuÞ;
Aj ¼
n X
A0ji ;
i¼1
M0ji ðtÞ ¼ N0ji ðtÞA0ji ðtÞ;
Mj ¼
n X
M0ji :
ð6Þ
i¼1
When censoring Scheme j is used, N0ji has a jump of size 1 at the observed failure time T0i of item i and Y0ji(t) indicates whether item i is in use (on test) at time t. In the following, we will assume one of the three schemes is used for the whole experiment. However, our results will hold if one combines the different schemes for the same experiment, for example, remove all but R*1 at t1 (Scheme 3) and then remove R2 items at t2 (Scheme 2). The organization of this article is as follows: We state the main results in Section 2. In Section 3, we define Schemes 1–4 explicitly on the same probability space and prove a number of martingale properties that are used in our proofs of the main results. In Section 4, expressions for the bias of our estimators are obtained (Theorem 7) and consistency of our estimators is established (Theorem 8). A small simulation is conducted in Section 5 to demonstrate the applicability of the theory. Proofs of results on bias, as well as Theorems 1 and 2, are given in the Appendix (Section Appendix A). 2. Main results The main result of this paper is Theorem 1 which states that the censoring schemes can be defined on the Rt same probability space such that M j ¼ N 0j 0 Y 0j ðuÞdL0 ðuÞ, j = 1,2,3,4 satisfy supt jMj ðtÞM4u ðtÞj ¼ OP ðn1=4 Þ. As a consequence, one can deduce that the product limit estimators under the different schemes have the same asymptotic distribution (Theorem 2). We define the product limit estimators ( ) Y DN 0j ðuÞ ^S j ðtÞ ¼ 1 , j ¼ 1,2,3,4, ð7Þ Y 0j ðuÞ urt where DN 0j ðuÞ ¼ N 0j ðuÞN 0j ðuÞ is the number of T0i that are observed to occur at time u. Since the failure time distribution is continuous, this number is either 0 or 1. We also define Nelson–Aalen estimators, j= 1,2,3,4, for the common L0 ðtÞ by Z t n X dN 0j ðuÞ ^ ðtÞ ¼ L I½Ti0 rt,Ti0 rC0i 4Cji ðY 0j ðTi0 ÞÞ1 : ð8Þ ¼ j Y ðuÞ 0 0j i¼1 We will assume, according to which scheme is used: (A1) 0 rp1 ,p2 , . . . ,pm o1. P (A2) r m m j ¼ 1 r‘ o 1, where r‘ ¼ r‘u=Sðt‘ Þ,‘ ¼ 1, . . . ,m. (A3) 0 or1 r Sðt1 Þ; 0 or‘ =r‘1 r Sðt‘ Þ=Sðt‘1 Þ,‘ ¼ 2, . . . ,m: Note that assumptions (A2) and (A3) depend on the unknown S, while (A1) does not. Asymptotic equivalence of Schemes 1 and 2 (Theorem 1) will hold when p‘ ¼ r‘ ð1r ‘1 Þ1 , 2 r‘ rm, ð9Þ Q‘1 or equivalently, when r‘ ¼ p‘ j ¼ 1 ð1pj Þ. Condition (9) equates the expected proportions of items removed for Schemes 1 and 2. For Scheme 3, we will let (equating expected proportions) p1 ¼ r1 ,
ð1p1 Þ ¼
r1 ; Sðt1 Þ
ð1p‘ Þ ¼
r‘ Sðt‘1 Þ Sðt Þ , ‘ ¼ 2, . . . ,m, r‘1 ‘
ð10Þ
Q that is, when r‘ ¼ Sðt‘ Þ ‘k ¼ 1 ð1pk Þ. If (9) and (10) hold, assumptions (A1)–(A3) are equivalent. The following theorem shows that the three schemes are asymptotically equivalent. Recall the notation given by (6). Theorem 1. Suppose that (Aj) holds for censoring Scheme j , j ¼ 1,2,3. One can define a probability space, on which T0i , Cji, C4i, i ¼ 1,2, . . . ,n, and filtration F n ðtÞ are defined, such that C4i are iid with distribution function (4), D j ðtÞ ¼ M j ðtÞM 4 ðtÞ is an
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
913
F n ðtÞ-martingale and sup jD j ðtÞj ¼ OP ðn1=4 Þ,
ð11Þ
0 r t r tF
Y ðtÞ Y ðtÞ 0j 04 sup ¼ a:s: Oðn1=2 ðloglognÞ1=2 Þ: n 0 r t r tF n
ð12Þ
Under (A2), p‘ is defined by (9). Under (A3), p‘ is defined by (10). The construction and martingale properties in the statement of Theorem 1 are detailed in Section 3. The convergence statements (11) and (12) are proven in the Appendix (Section Appendix A). As a consequence of Theorem 1, we obtain the following weak convergence result for progressively censored data. Theorem 2. Let T10 ,T20 , . . . ,Tn0 be independent random variables each with continuous distribution function F0 and subject to iid censoring by C01 , . . . ,C0n , and progressive censoring under Scheme j, j ¼ 1,2,3. Suppose assumption (Aj) holds for Scheme j. Then, for any t satisfying SðtÞ 4 0, pffiffiffi nðF^ j F0 Þ-D ð1F0 ÞWðvÞ, Rs j ¼ 1,2,3, in the space D[0,t], as n-1, where W is a Wiener process and vðsÞ ¼ 0 ðpðuÞÞ1 dL0 ðuÞ, with p defined by (5) and the p‘ , given by (9) for j =2, resp. (10) for j =3. Also, for j =1,2,3, pffiffiffi ^ nðF j F0 Þ -D WðvÞ: ð1F^ j Þ The proof of Theorem 2 is given in the Appendix (Section Appendix A). While the experimenter knows the values of pk, Rk and R*k (1 rk r m), the distribution functions F0 and G0, L0 and v are all unknown. Uniform confidence bands can be obtained directly using Theorem 2 and Z s ððY 0j ðuÞ1ÞY 0j ðuÞÞ1 dN 0j ðuÞ, j ¼ 1,2,3, v^ j ðsÞ ¼ n 0
to estimate v. In the iid random censorship case, such bands were given by Gill (1980). See also the related bands (Gillespie ¨ o+ and Horva´th, 1986; Burke et al., 1981, p. 107). One can also obtain the following Hall–Wellnerand Fisher, 1979; Csorg type bands (Hall and Wellner, 1980), which were shown to have good moderate sample size properties. Corollary 3. Under the conditions of Theorem 2, n1=2 ðF^ j F0 Þ ð1 þ v^ j Þð1F^ j Þ
-D W o ðK0 Þ,
j =1,2,3, as n-1, on D[0,t], where K0 ðsÞ ¼ vðsÞ=ð1 þvðsÞÞ and Wo is a Brownian bridge process on [0,1]. Hence, for a ¼ K0 ðtÞ, o ^ j ðsÞÞ1 ðS^ j ðsÞÞÞ1 S^ j ðsÞ 7n1=2 cð1 aÞ ðð1 þ v
ð13Þ
o are uniform (s rt) confidence bands for S0 ¼ 1F0 , where j ¼ 1,2,3, where P½sup0 r x r a jW o ðxÞj r cð1 aÞ ¼ 1a.
A simulation is conducted in Section 5 using (13). One can also obtain confidence bands for the cumulative hazard function, based on the Nelson–Aalen estimator (Andersen et al., 1993, Sections 4.1 and 4.2). 3. Construction and some martingale results Let Ni0 ðtÞ ¼ I½Ti0 rt,
A0i ðtÞ ¼
Z
t 0
I½u r Ti0 dL0 ðuÞ,
Mi0 ¼ Ni0 A0i ,
ð14Þ
i= 1,2,y,n. Although not observable under the censoring schemes, these processes are the elementary objects that are used to prove the martingale properties of the censored processes. It is well known that M0i is a martingale with respect to its natural filtration sfI½Ti0 r u : u rtg and with respect to F ðnÞ ðtÞ ¼ sfI½Ti0 r u : u rt, i ¼ 1,2, . . . ,ng:
ð15Þ F n ðtÞ
of (17), which contains It is shown (Theorem 4) that it is also a martingale with respect to the enlarged filtration F ðnÞ ðtÞ, the censoring information, up to time t, by the C0i, the three progressive censoring schemes and a random right censoring scheme using iid random variables all defined below. Some consequences are that the corresponding processes M0ji defined by (6) for each of the four censoring schemes are martingales (Corollary 5). Moreover, the differences, M0jui M04i , ju ¼ 1,2,3 are martingales with respect to F n ðtÞ ( Corollary 6). This is used in Appendix A to prove asymptotic equivalence of the schemes (proof of Theorem 1).
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We will define the three progressive censoring schemes explicitly as follows: Let fUiðkÞ : k ¼ 1,2, . . . ,m; i ¼ 1,2, . . . ,ng be a rectangular array of independent uniform (0,1) random variables, independent of the Ti ¼ Ti0 4C0i , i ¼ 1,2, . . . ,n. We will define (risk) sets of indices rj ðtk , oÞ and censoring variables Cji ðoÞ, recursively, for each Scheme j, time tk, and item i, where j ¼ 1,2,3; k ¼ 1,2, . . . ,m; i ¼ 1,2, . . . ,n. Put rj ðt1 , oÞ ¼ fi : t1 rTi ðoÞg for each j ¼ 1,2,3. Notation: njk #rj ðtk , oÞ ¼ Y 0j ðtk Þ;
mjk #rj ðtk þ, oÞ ¼ Y 0j ðtk þÞ:
ð16Þ ð1Þ Uið1Þ r Uðbn , 11 p1 cÞ
ð1Þ Uðbn 11 p1 cÞ
where is Scheme j =1: (Progressive Type-p). Item with index i 2 r1 ðt1 , oÞ is removed at time t1, if the bn11 p1 c-order statistic from the set fUið1Þ : i 2 r1 ðt1 , oÞg, and put C1i ¼ t1 , for this i. Let r1 ðt2 , oÞ ¼ fi : t2 rTi ðoÞ,C1i ðoÞat1 g denote the risk set of the items at time t2. ðkÞ We continue in this way for k ¼ 2, . . . ,m, where item with index i 2 r1 ðtk , oÞ is removed at time tk, if UiðkÞ rUðbn , where 1k pk cÞ ðkÞ Uðbn 1k pk cÞ
is the bn1k pk c-order statistic from the set fUiðkÞ : i 2 r1 ðtk , oÞg, and put C1i ¼ tk . We define r1 ðtk þ 1 , oÞ ¼ fi : tk þ 1 r
Ti ðoÞ,C1i ðoÞ tk g, for k rm1. Finally, put C1i ðoÞ ¼ tF , if item i has not been removed at times t1 , . . . ,tm under Scheme 1. Scheme j =2: (Progressive Type-R). For k ¼ 1,2 . . . ,m, recursively, item with index i 2 r2 ðtk , oÞ is removed at time tk, if ðkÞ ðkÞ UiðkÞ r UðR , where UðR is the Rk-th order statistic from the set fUiðkÞ : i 2 r2 ðtk , oÞg. Put C2i ¼ tk . We define r2 ðtk þ 1 , oÞ ¼ kÞ kÞ
fi : tk þ 1 r Ti ðoÞ, C2i ðoÞ tk g, for kr m1. Put C2i ðoÞ ¼ tF , if item i has not been removed at times t1 , . . . ,tm under Scheme 2. ðkÞ Scheme j ¼ 3: (Progressive Type-R*). For k ¼ 1,2 . . . ,m, recursively, item i 2 r3 ðtk , oÞ is removed at time tk, if UiðkÞ r Uððn
þ jk Rk Þ
where
ðkÞ Uððn þ jk Rk Þ Þ
is the ðnjk Rk Þ þ th order statistic from the set
fUiðkÞ
Þ
,
: i 2 r3 ðtk , oÞg. Put C3i ¼ tk . Define r3 ðtk þ 1 , oÞ ¼
fi : tk þ 1 r Ti ðoÞ, C3i ðoÞ tk g, for k rm1. Put C3i ðoÞ ¼ tF , if item i has not been removed at times t1 , . . . ,tm under Scheme 3. Then, N0ji ðtÞ ¼ I½Ti0 r t,Ti0 r C0i 4Cji and Y0ji ðtÞ ¼ I½Ti 4Cji Zt, i ¼ 1,2, . . . ,n, have the required joint distribution for Scheme j, j ¼ 1,2,3. For these types of censoring, it is important to note two things: Firstly, events of the form ½Cji ¼ tk ,tk oTi depend on the risk set rj ðtk , oÞ, but do not depend on the future values of the Tiu , namely Tiu I½Tiu 4tk , iu ¼ 1,2, . . . ,n. 0
That is, ½Cji ¼ tk ,tk oTi 2 sfI½Tiu rtku ,Uiðk0 Þ ,1 r ku r k,1 r iu r ng: Secondly, for tk os, P½s o Ti0 jtk oTi ,Cji ¼ tk ¼ P½s oTi0 jtk oTi ¼ P½s o Ti0
jtk o Ti0 ,
that is, knowing that item i would be censored at tk, under Scheme j, does not affect the conditional failure time distribution of T0i , nor does it change the conditional failure time distribution of those functioning items that are not removed. We will compare these progressive censoring schemes with the iid scheme. Scheme 4 (IID random) For the same array U(k) i , 1 r kr m, i ¼ 1,2, . . . ,n, define the random variables C4i as 8 ð1Þ > t1 if Ui r p1 , > > < ðkÞ ð‘Þ C4i ¼ t‘ if pk o Ui for k o‘ and Ui r p‘ , > > ð‘Þ > :t if p oU for all ‘ r m: F
‘
i
The C4i have distribution (4) and are independent of the failure times Ti. Let F 3G denote the smallest s-field, containing the sets in F and G. With the Cji as defined in this Section and using the notation (6), define, for i ¼ 1,2, . . . ,n and j ¼ 1, 2, 3, 4, F ji ðtÞ ¼ sfN0ji ðuÞ,I½Ti 4Cji r u,Cji o Ti , I½Ti 4Cji ru,C0i oTi0 4Cji : 0 ru rtg, and F nðjÞ ðtÞ ¼
Wn
i¼1
F ji ðtÞ, the smallest s-field containing all the events in F ji ðtÞ, 1r i rn. The filtration F ðjÞ n ðtÞ is right
continuous and M j ðtÞ is adapted to it. In order to compare the processes under different censoring schemes, we will consider the larger filtration: 4 _ ðF nðjÞ ðtÞÞ, ð17Þ F n ðtÞ ¼ F ðnÞ ðtÞ3 j¼1
where F ðnÞ ðtÞ is defined by (15). Eq. (17) contains all the information concerning which items are censored up to time t as well as the failure times of the items, whether censored by one or more of the schemes or not, up to time t. Consider the following thought experiment. Instead of removing a functioning item at time tk when it is censored, place a card beside it indicating the time tk and the scheme (j = 0, 1, 2, 3 or 4) under which it is censored. Each item may have one or more (up to five) cards placed beside it. In this way, its lifetime after censoring can still be observed. The filtration F n ðtÞ contains all this information regarding failure times, up to time t, and which items are carded and by which scheme, up to time t. These remarks and the following theorem show that Schemes 1–3 satisfy the general definition of independent censoring (Gill, 1980; Andersen et al., 1993, p. 139). Theorem 4. The processes M0i of (14) and variation process /Mi0 ,Mi0 SðtÞ ¼ A0i ðtÞ;
Pn
i¼1
Mi0 are martingales with respect to the filtration F n ðtÞ. M0i has quadratic
/Mi0 ,Mi00 SðtÞ ¼ 0, iaiu:
ð18Þ
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
915
Corollary 5. The processes M0ji(t) of (6) are martingales with respect to F n ðtÞ and hence with respect to the filtration F ðjÞ n ðtÞ with quadratic variation process /M0ji ,M0ji SðtÞ ¼ A0ji ðtÞ;
/M0ji ,M0jiu SðtÞ ¼ 0, iaiu,
where A0ji(t) is defined by (6). Moreover, M j is an F n ðtÞ-martingale with quadratic variation process /M j ,M j SðtÞ ¼ A j ðtÞ. Corollary 6. The processes Djui ðtÞ ¼ M0jui ðtÞM04i ðtÞ, are zero-mean
ju ¼ 1,2,3, i ¼ 1,2, . . . ,n,
F n ðtÞ-martingales
/Djui ,Djui SðtÞ ¼
Z
t
ð19Þ
with quadratic variation process
ðY0jui Y04i Þ2 dL0 ;
/Djui ,Djuiu SðtÞ ¼ 0 for iaiu:
0
Moreover, D ju
Pn
i¼1
Djui is an F n ðtÞ-martingale having quadratic variation process /D ju ,D ju SðtÞ ¼
Pn
i¼1
Rt
2 0 ðY0jui Y04i Þ dL0 .
Proof of Theorem 4. Clearly, M0i (t) is adapted to the filtration F n ðtÞ. The sfield F n ðtÞ is generated by sets of the form T T G ¼ ni0 ¼ 1 ð 4j ¼ 1 Bjiu \ B0jiu \ Jiu Þ, where Jiu ¼ ½Ti00 ru, for some u r t, or ½t oTi00 , B0jiu ¼ ½Tiu 4Cjiu r u,C0iu oTi00 4Cjiu , for some u rt, or their complements, and Bjiu ¼ ½Tiu 4Cjiu r u, Cjiu o Tiu , for some u rt, or their complements. We split the following integral into two parts Z G
Mi0 ðtuÞMi0 ðtÞ dP ¼
Z
Z þ
G\½Ti0 r t
G\½t o Ti0
fMi0 ðtuÞMi0 ðtÞg dP:
ð20Þ
The first term on the right side of (20) equals 0, since IG\½T 0 r t ðMi0 ðtuÞMi0 ðtÞÞ ¼ 0, a:s: For the second term of (20), let i T T Giu ¼ iuai ð 4j ¼ 1 Bjiu \ B0jiu \ Jiu Þ \ Bji \ B0ji . Then, Z Z Z Ni0 ðtuÞNi0 ðtÞ dP ¼ IGui I½t oTi0 fNi0 ðtuÞNi0 ðtÞg dP ¼ PðGuÞ I½t oTi0 fNi0 ðtuÞNi0 ðtÞg dP ¼ PðGiuÞ½F0 ðtuÞF0 ðtÞ: ð21Þ i G\½t o Ti0
The second equality follows because T0i and the Ti00 are independent and the five censoring mechanisms, up to time t, do not depend on the value of T0i when ½t o Ti0 . Note also that ½Ti 4Cji ru,Cji o Ti , for some u rt \ ½t oTi0 ¼ ½Cji r t,Cji o C0i \ ½t o Ti0 and ½Ti 4Cji r u,C0i oTi0 4Cji ,for someu r t \ ½t o Ti0 ¼ ½C0i rt,C0i o Cji \ ½t o Ti0 . We also have Z G\½t o Ti0
Z
t0 t
I½u r Ti0 dL0 ðuÞ dP ¼
Z
IGiu
¼ PðGiuÞ
Z
t0
t
Z
I½t oTi0 I½u rTi0 dL0 ðuÞ dP ¼
Z
IGiu
Z t
t0
I½u r Ti0 dL0 ðuÞ dP
t0
ð1F0 ðuÞÞ dL0 ðuÞ dP, t
which equals the right side of (21). Hence the second term of (20) equals zero. Since the sets G form a p-system which generates F n ðtÞ, E½Mi0 ðtuÞMi0 ðtÞjF n ðtÞ ¼ 0, almost surely, (Billingsley, 1995, Theorem 34.1). Consequently, Pn M0i (t) is a martingale. The sum of martingales, i ¼ 1 Mi ðtÞ, with respect to the same filtration, is clearly a martingale. Consider the vector ðN1 ,N2 , . . . ,Nn Þ. Since each Ni is a counting process and the jumps of any two of them cannot occur simultaneously, with probability 1, this vector is a multivariate counting process. Hence (Gill, 1980, Appendix 1) Z t Z t ð1DAi ÞdAi ; /Mi ,Miu SðtÞ ¼ DAi dAiu , iaiu /Mi ,Mi SðtÞ ¼ 0
0
Since the distribution F0 is assumed to be continuous, (18) holds and hence the theorem.
&
Proof of Corollary 5. Since the process Y0ji(t) is measurable with respect to F nðjÞ ðtÞ D F n ðtÞ and left continuous, it is Rt 0 predictable for both s-fields. Consequently, M0ji ðtÞ ¼ 0 Y0ji ðuÞ dM i ðuÞ are F n ðtÞ-martingales. Moreover, since M0ji(t) is ðjÞ F ðjÞ n ðtÞ-measurable, each M0ji(t) is an F n ðtÞ-martingale. From stochastic integration theory Z Z Z Y0ji Y0jiu d Mi ,Miu Y0ji ðuÞ dMi ðuÞ, Y0jiu ðuÞ dMiu ðuÞ ¼
and the second statement of Corollary 5 follows from (18).
&
The proof of Corollary 6 is similar, since Y0jui ðtÞY04i ðtÞ is left continuous and F n ðtÞ-measurable implies that Rt 0 Djui ðtÞ ¼ 0 ðY0jui Y04i Þ dMi is an F n ðtÞ-martingale.
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M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
^ 4. Bias and consistency of S^ j and L j Having established that M0ji and M j of (6) are martingales, we can use standard martingale arguments to derive properties of our estimators. Theorem 7 concerns a bound on the bias of S^ j , j ¼ 1,2,3. Let t0 ¼ 0, tm þ 1 ¼ tF and p0 ¼ 0. Define, for k ¼ 1,2, . . . ,m, hk ðuÞ ¼ ðSðtk ÞSðuÞÞ=Sðtk Þ,
u Z0,
q1k ðxÞ ¼ hk1 ðtk Þ þ xð1pk Þ Sðtk Þ=Sðtk1 Þ,
q3k ðxÞ ¼ hk1 ðtk Þ þ xrk Sðtk Þ=Sðtk1 Þ,
x Z0,
x Z 0:
ð22Þ
Theorem 7. If SðtÞ 40 and (A1) holds for Scheme 1, (resp. (A2) holds for Scheme 2 and (A3) holds for Scheme 3), then 0 r ES^ j ðtÞS0 ðtÞ r f1S0 ðtÞgP½Y 0j ðtÞ ¼ 0 r f1S0 ðtÞgfgj ðtÞgn ,
ð24Þ
and gj ðtÞ o 1, where gj is defined by ( 1SðtÞ g1 ðtÞ ¼ q11 3q12 3 3q1k 3hk ðtÞ
g2 ðtÞ ¼
k Y
FðnðtÞÞ
g3 ðtÞ ¼
if t rt1 if tk o t rtk þ 1 ;
Sðtw Þ )rw ! Sðtw Þ o1; FðnðtÞÞFðtw Þ
ð25Þ
(
w¼1
(
ð23Þ
1SðtÞ
if t rt1
q31 3q32 3 3q3k 3hk ðtÞ
if tk o t rtk þ 1 ;
ð26Þ
ð27Þ
Hence, S^ j ðtÞ is an asymptotically unbiased estimator of S(t) with bias converging to zero at an exponential rate as n-1. Theorem 7 is well known when iid censoring is used (Fleming and Harrington, 1991, Lemma 3.2.1). In this case, P½Y 04 ðuÞ ¼ 0 ¼ fpðuÞgn , where p is defined by (5). The proof of Theorem 7 is in the Appendix. Consider I½Y 0j ðtÞ 40=Y 0j ðtÞ, which is taken to be zero if the numerator is zero, j =1,2,3,4. Because of left continuity, I½Y 0j ðtÞ 40=Y 0j ðtÞ is predictable with respect to F ðjÞ n ðtÞ. Consequently, for the Nelson–Aalen estimators (8), on letting R Lj ðtÞ ¼ 0t I½Y 0j ðuÞ 4 0 dL0 ðuÞ, ^ ðtÞL ðtÞ ¼ L j j
Z
t 0
I½Y 0j ðuÞ 4 0ðY 0j ðuÞÞ1 dM j ðuÞ
^ ^ is a well-defined stochastic integral and a F ðjÞ n ðtÞmartingale. Hence, EfL j ðtÞLj ðtÞg ¼ 0, and the bias EfL j ðtÞLðtÞg ¼ Rt ^ ðtÞ is an asymptotically unbiased estimator of 0 P½Y 0j ðuÞ ¼ 0dL0 ðuÞ. Using (24), it follows that for Schemes ju ¼ 1,2,3, L ju LðtÞ, with bias converging to zero at an exponential rate, as n-1, for t 2 ½0, tF Þ, under the conditions of Theorem 7. The following establishes weak consistency for the Nelson–Aalen and product limit estimators under progressive censoring Schemes 1–3.
Theorem 8. (a) If t 2 ð0, tF satisfies Y 0j ðtÞ-P 1
as n-1,
ð28Þ
then ^ ðsÞL0 ðsÞj-P 0, sup jL j
as n-1;
ð29Þ
0rsrt
sup jF^ j ðsÞF0 ðsÞj-P 0
as n-1,
ð30Þ
0rsrt
^ j , is defined by (7), resp. (8). j = 1,2,3, where F^ j ¼ 1S^ j , and S^ j , resp. L (b) If x 2 ð0, tF is such that (28) holds for all t ox, then (30) holds with t replaced by x. Using the martingale properties derived in Section 3 and Theorem 7, the proof of Theorem 8 is similar to that of Theorem 3.4.2 of Fleming and Harrington (1991). Note that condition (28) will hold under the conditions of Theorem 1.
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
917
5. Simulation To explore the applicability of our results, a simulation was conducted based on 10,000 replications. Simulated data (T0i ) from a Weibull distribution with scale 10 and shape 2 were generated for sample sizes 100 and 200. These data were subjected to random right censoring by C0i having distribution G0 as Weibull with scale 10/9 and shape 2, giving about a 10% censorship rate. Then, the three progressive censoring schemes were applied to the Ti ¼ minðTi0 ,C0i Þ according to the construction in Section 3 at time points t = 2.5, 5 and 9. We specified the number of items to be removed under Scheme 2, namely R ¼ ðR1 ,R2 ,R3 Þ. Then, using Eqs. (9) and (10), we record the corresponding values of p ¼ ðp1 ,p2 ,p3 Þ and R ¼ ðR1 ,R2 ,R3 Þ for Schemes 1 and 3, respectively. A comparison to random right censoring (Scheme 4 with the same p) is also given. In each case, uniform confidence bands were calculated using (13). The percent reduction in the total time on test (TTT) is also given for each scheme. The results are summarized in Tables 1 and 2 for n = 100 and 200, respectively. In Table 1, we chose R to be (40,5,5), (5,40,5), (5,30,15) and (17,17,16) in order to study the effects of censoring either early, later or more evenly. Note that (5,30,15) was chosen over (5,5,40) because the latter resulted in too few items on trial after t= 9 for the asymptotic theory to hold. As can be seen, the coverage probabilities were all close to the nominal value of 95%. The average TTT if no progressive censoring took place was 781.2. The percent total time on test reduction was the greatest when R= (40,5,5) (early removal) and the least when R= (5,15,30), where 30 items were removed at t =9. This is consistent with the ‘‘new is better than used’’ property of the Weibull. The average band widths (W) together with the standard deviation of the widths (in parentheses) are given. W at t = 2 was 0.264 for each scheme and each value of R, since progressive censoring started after this time. The W values are quite comparable for each scheme. For example, when Scheme 2 with R ¼ ð40,5,5Þ is used, the widths of the Hall–Wellner confidence bands at times t =2, 4, 6.5, 9.5 were 0.264, 0.281, 0.330, 0.405, respectively. The coverage probability was 95.9%. The average TTT was 503.1 (a 35.6% reduction from 781.2). If we compare these results with a sample size of n= 50 to start and no progressive censoring, the widths would be have been 0.328, 0.375, 0.375, 0.376 at times t= 2, 4, 6.5, 9.5, respectively, with a coverage probability of 95.0%. The average TTT would have been 418.4 (smaller than 503.1). However, much narrower widths, at t = 2, 4, 6.5, were achieved by the progressive censoring of 50 items from 100, than one would get with an original sample size of 50. The larger width at t =9.5 was the result of very few items on test after t= 9. In Table 2, 100 items were progressively censored from n ¼ 200 in the same proportions as Table 1. As can be seen, the overall results are similar to those of Table 1. The average total time on test if no progressive censoring took place was 1746.5. For example, Scheme 2 with R= (80,10,10) (a total of 100 items removed), saw an average TTT of 1124.4 (35.6% reduction). The widths of the Hall–Wellner confidence bands at times t = 2, 4, 6.5, 9.5 were 0.191, 0.202, 0.240, 0.302, respectively. The coverage probability was 96.1%. If we compare the results with a sample size of n= 100 to start and no
Table 1 Simulation from a Weibull distribution (scale 10; shape 2) and progressive censoring at t = 2.5, 5 and 9. Various choices for R are specified. Average coverage probabilities (C.P. in %), reduction of the total time on test (%), and widths, W, (with std. dev.) of uniform confidence bands given by (13) at t= 2, 4, 6.5, 9.5 under the three progressive schemes and Scheme 4: iid random censoring. n =100
Scheme 1
Scheme 2
Scheme 3
Scheme 4
C.P. (%) TTT red. (%) W at 2 4 6.5 9.5
p ¼ ð0:43,0:12,0:24Þ 95.9 34.8 0.264 (0.036) 0.281 (0.008) 0.328 (0.015) 0.400 (0.022)
R ¼ ð40,5,5Þ 95.9 35.6 0.264 (0.036) 0.281 (0.008) 0.330 (0.017) 0.405 (0.026)
R ¼ ð54,39,16Þ 96.0 34.9 0.264 (0.036) 0.281 (0.008) 0.328 (0.015) 0.402 (0.026)
p ¼ ð0:43,0:12,0:24Þ 95.8 35.6 0.264 (0.036) 0.281 (0.009) 0.331 (0.021) 0.408 (0.036)
C.P. (%) TTT red. (%) W at 4 6.5 9.5
p ¼ ð0:05,0:56,0:29Þ 95.9 29.3 0.270 (0.001) 0.312 (0.023) 0.424 (0.036)
R ¼ ð5,40,5Þ 96.1 30.1 0.270 (0.001) 0.314 (0.026) 0.429 (0.043)
R ¼ ð89,32,13Þ 95.7 29.3 0.270 (0.001) 0.312 (0.024) 0.427 (0.041)
p ¼ ð0:05,0:56,0:29Þ 95.6 30.0 0.270 (0.001) 0.314 (0.028) 0.431 (0.053)
C.P. (%) TTT red. (%) W at 4 6.5 9.5
p ¼ ð0:05,0:42,0:67Þ 94.5 27.7 0.270 (0.001) 0.297 (0.012) 0.363 (0.052)
R ¼ ð5,30,15Þ 95.2 28.5 0.270 (0.001) 0.298 (0.013) 0.362 (0.058)
R ¼ ð89,42,8Þ 94.7 27.8 0.270 (0.001) 0.297 (0.013) 0.366 (0.054)
p ¼ ð0:05,0:42,0:67Þ 93.9 28.4 0.270 (0.001) 0.298 (0.015) 0.365 (0.061)
C.P. (%) TTT red. (%) W at 4 6.5 9.5
p ¼ ð0:18,0:26,0:69Þ 94.4 29.3 0.273 (0.002) 0.299 (0.009) 0.351 (0.048)
R ¼ ð17,16,17Þ 94.9 30.1 0.273 (0.002) 0.300 (0.010) 0.350 (0.053)
R ¼ ð77,46,8Þ 94.6 29.6 0.273 (0.002) 0.299 (0.010) 0.354 (0.054)
p ¼ ð0:18,0:26,0:69Þ 94.0 30.0 0.273 (0.002) 0.300 (0.012) 0.352 (0.057)
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M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
Table 2 Simulation from a Weibull distribution (scale 10; shape 2) and progressive censoring at t = 2.5, 5 and 9. Various choices for R are specified. Average coverage probabilities (C.P. in %), reduction of the total time on test (%), and widths, W (with std. dev.), of the uniform confidence bands given by (13) at t = 2, 4, 6.5, 9.5 under the three progressive schemes and Scheme 4: iid random censoring. n= 200
Scheme 1
Scheme 2
Scheme 3
Scheme 4
C.P. (%) TTT red. (%) W at 2. 4. 6.5 9.5
p ¼ ð0:43,0:12,0:24Þ 96.2 35.2 0.191 (0.003) 0.202 (0.004) 0.239 (0.008) 0.300 (0.014)
R ¼ ð80,10,10Þ 96.1 35.6 0.191 (0.003) 0.202 (0.004) 0.240 (0.009) 0.302 (0.016)
R ¼ ð107,77,32Þ 96.0 35.2 0.191 (0.003) 0.202 (0.004) 0.239 (0.008) 0.300 (0.015)
p ¼ ð0:43,0:12,0:24Þ 96.2 35.6 0.191 (0.003) 0.202 (0.004) 0.240 (0.011) 0.303 (0.021)
C.P. (%) TTT red. (%) W at 4. 6.5 9.5
p ¼ ð0:05,0:56,0:29Þ 96.6 29.7 0.192 (0.000) 0.230 (0.012) 0.322 (0.023)
R ¼ ð10,80,10Þ 96.6 30.1 0.192 (0.000) 0.231 (0.013) 0.326 (0.028)
R ¼ ð177,64,25Þ 96.3 29.6 0.192 (0.000) 0.230 (0.013) 0.323 (0.024)
p ¼ ð0:05,0:56,0:29Þ 96.6 30.1 0.192 (0.000) 0.231 (0.014) 0.327 (0.032)
C. P. (%) TTT red. (%) W at 4. 6.5 9.5
p ¼ ð0:05,42,0:67Þ 95.7 28.1 0.192 (0.000) 0.215 (0.006) 0.283 (0.042)
R ¼ ð10,60,30Þ 94.8 28.5 0.192 (0.000) 0.216 (0.007) 0.289 (0.051)
R ¼ ð177,84,15Þ 95.5 28.2 0.192 (0.000) 0.215 (0.007) 0.283 (0.044)
p ¼ ð0:05,42,0:67Þ 95.4 28.5 0.192 (0.000) 0.216 (0.008) 0.285 (0.049)
C.P. (%) TTT red. (%) W at 4 6.5 9.5
p ¼ ð0:18,0:27,0:68Þ 95.5 30.0 0.194 (0.001) 0.216 (0.005) 0.274 (0.038)
R ¼ ð33,34,33Þ 94.6 30.1 0.195 (0.001) 0.217 (0.005) 0.273 (0.046)
R ¼ ð154,91,16Þ 95.2 29.8 0.194 (0.001) 0.216 (0.005) 0.274 (0.041)
p ¼ ð0:18,0:27,0:68Þ 95.2 30.1 0.195 (0.001) 0.217 (0.006) 0.275 (0.046)
progressive censoring (not shown in the tables), the widths would be have been 0.264, 0.269, 0.270, 0.273 at times t =2, 4, 6.5, 9.5, respectively. with a coverage probability of 95.7%. The average TTT was 781.2. Thus, much narrower widths were achieved by starting with 200 items on trial and the progressive censoring of 100 items of them, than one would get with an original sample size of 100, especially at smaller values of t. Thus, using progressive censoring for 200 items is a way to obtain narrower confidence bands at smaller t (as compared with starting with 100 items). The TTT is controlled by the systematic removal of items (1124.4 versus 1746.5), but is larger than the TTT of starting with 100 items and no progressive censoring (781.2).
Acknowledgements The author is grateful to the Executive and Associate Editors and to the referees for their insightful and helpful comments, which led to a substantial improvement of this paper. Appendix A. Proofs A.1. Proof of Theorem 7 Rt ^ ðuÞ, where As in the iid censoring case, the product limit estimator (7) can be derived recursively: S^ j ðtÞ ¼ 1 0 S^ j ðuÞ dL j ^ L j is defined by (8), and a bound on the bias can be found for Schemes 1–3. If S0 ðtÞ 40, then (Fleming and Harrington, 1991, Theorem 3.2.3 and Corollary 3.2.1) ( ) Z t^ S^ j ðtÞ S j ðuÞ dN 0j ðuÞ ¼ 1 dL0 ðuÞ , S0 ðtÞ Y 0j ðuÞ 0 S0 ðuÞ S^ j ðtÞS0 ðtÞ ¼ S0 ðtÞ
Z
t 0
Hj ðuÞ dM j ðuÞ þBj ðtÞ,
ð31Þ
where Hj ðuÞ ¼ S^ j ðuÞI½Y 0j ðuÞ 4 0=ðS0 ðuÞ Y 0j ðuÞÞ and Bj ðtÞ ¼ I½tjn o t
S^ j ðtjn ÞfS0 ðtjn ÞS0 ðtÞg , S0 ðtjn Þ
ð32Þ
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
919
and tjn ¼ inffu : Y 0j ðuÞ ¼ 0g, j ¼ 1,2,3. The integral on the right side of (31) is a martingale, since the integrand is F ðjÞ n ðuÞ-predictable. Hence, it has expectation zero and (23) follows. To prove (24), we let Ej ¼ Ej ðtÞ ¼ ½Y 0j ðtÞ ¼ 0, j ¼ 1,2,3. If t r t1 , then none of the items are subject to censoring by P Scheme 1 before time t. Clearly, PðEj Þ ¼ f1SðtÞgn , j ¼ 1,2,3. Let N j ðuÞ ¼ ni¼ 1 I½Ti ru,Ti rCji . For t o tF , tk o t rtk þ 1 , 1 rk rm and mjk given by (16), P½Ej ¼
mj1 n X X ‘1 ¼ 0 ‘2 ¼ 0
mj,k1
X
P½N j ðt1 Þ ¼ ‘1 P½N j ðt2 Þ ¼ ‘ 2 jN j ðt1 Þ ¼ ‘1 P½N j ðtk Þ
‘k ¼ 0
¼ ‘ k jN j ðt1 Þ ¼ ‘1 , . . . ,N j ðtk1 Þ ¼ ‘ k1 P½Ej jN j ðt1 Þ ¼ ‘1 , . . . ,N j ðtk Þ ¼ ‘ k , P where ‘ k ¼ kw ¼ 1 ‘w and, with hw defined by (22), ! mj,k1 ðhk1 ðtk ÞÞ‘k ðSðtk Þ=Sðtk1 ÞÞmj,k1 ‘k ; P½N j ðtk Þ ¼ ‘ k jN j ðt1 Þ ¼ ‘1 , . . . , N j ðtk1 Þ ¼ ‘ k1 ¼ ‘k P½Ej jN j ðt1 Þ ¼ ‘1 , . . . ,N j ðtk Þ ¼ ‘ k ¼ ðhk ðtÞÞmjk :
ð33Þ
ð34Þ
Scheme j= 1: We use (33) and (34) with m1w ¼ dn1w ð1pw Þe and n1w ¼ dn1,w1 ð1pw1 Þe‘w and obtain P½E1 jN 1 ðt1 Þ ¼ ‘1 , . . . ,N 1 ðtk1 Þ ¼ ‘ k1 ¼
dn1,k1X ð1pk1 Þe
P½N 1 ðtk Þ ¼ ‘ k jN 1 ðt1 Þ ¼ ‘1 , . . . ,N 1 ðtk1 Þ ¼ ‘ k1 P½E1 jN 1 ðt1 Þ ¼ ‘1 , . . . ,N 1 ðtk Þ ¼ ‘ k
‘k ¼ 0
¼ ðq1k ðhk ðtÞÞÞ
dn1,k1 ð1pk1 Þe
r ðq1k ðhk ðtÞÞÞ
n1,k1 ð1pk1 Þ
:
Similarly, P½E1 jN 1 ðt1 Þ ¼ ‘1 , . . . ,N 1 ðtk2 Þ ¼ ‘ k2 r
dn1,k2X ð1pk2 Þe
P½N 1 ðtk1 Þ ¼ ‘ k1 jN 1 ðt1 Þ ¼ ‘1 , . . . ,N 1 ðtk2 Þ ¼ ‘ k2 ðq1k ðhk ðtÞÞÞn1,k1 ð1pk1 Þ
‘k1 ¼ 0
r ðq1,k1 3q1k 3hk ðtÞÞÞn1,k2 ð1pk2 Þ :
Continuing in this way, Eq. (24) follows and 0 og1 ðtÞ o 1. Scheme j= 2: The following lemma will be needed: Lemma 9. For any 0 o d o Sðtw Þ, let 1Fðtw Þ Sðtw Þ , hw ðxÞ ¼ xð1 þ dÞ ðxFðtw ÞÞ w ¼ 1,2, . . . ,m, where Fðtw Þ ox o 1. Then, there exists 0 o ew ¼ ew ðdÞ o1, such that, on ½ew ,1, hw is strictly increasing and hw ðxÞ o 1. P We use (33) and (34) with m2w ¼ nR w ‘ w and R k ¼ kw ¼ 1 Rw and obtain þ mX 2,k1 m2,k1 Fðtk ÞFðtk1 Þ ‘k Sðtk Þ ðm2,k1 ‘k Þ FðtÞFðtk Þ ðm2,k1 Rk ‘k Þ P½E2 jN 2 ðt1 Þ ¼ ‘ 1 , . . . ,N 2 ðtk1 Þ ¼ ‘ k1 r Sðtk1 Þ Sðtk1 Þ Sðtk Þ ‘k ‘k ¼ 0 m2,k1 Rk FðtÞFðtk1 Þ Sðtk Þ r : Sðtk1 Þ FðtÞFðtk Þ Using (33), it follows that Rw k Y Sðtw Þ PðE2 Þ rFðtÞn : FðtÞFðtw Þ w¼1 By assumption (A2), r k o 1, implies that du ð1r k Þ=r k 40 and " Sw #rw !n k Y Sðtw Þ PðE2 Þ r FðtÞ1 þ du : FðtÞFðtw Þ w¼1 By Lemma 9, there is a nðtÞ ¼ maxft,F 1 ðe1 ðduÞÞ, . . . , F 1 ðek ðduÞÞg, such that PðE2 Þ r g2 ðtÞ o1, where g2 is defined by (26). Scheme j = 3: We use (33) and (34) with m3w ¼ Y 03 ðtw Þ4Rw . Recall R‘ ¼ dnr ‘ e. Define n1 ðk1 Þ ¼ ðnk1 Þ4R1 , and nj ðk1 ,k2 , . . . ,kj Þ ¼ ðnj1 ðk1 ,k2 , . . . ,kj1 Þkj Þ4Rj . Similar to the case j ¼ 1, for tk o t rtk þ 1 , but t o tF , ! mX 3,k1 m3,k1 P½E3 jN 3 ðt1 Þ ¼ ‘1 , . . . ,N 3 ðtk1 Þ ¼ ‘ k1 ¼ ðhk1 ðtk ÞÞ‘k ðSðtk Þ=Sðtk1 ÞÞm3,k1 ‘k ðhk ðtÞÞm3k : ‘k ‘k ¼ 0
920
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
Note that rk ðm3,k1 ‘k Þ r m3,k and 0 o hk ðtÞ o 1, imply ðhk ðtÞÞm3k rðhk ðtÞÞrk ðm3,k1 ‘k Þ Hence, P½E3 jN 3 ðt1 Þ ¼ ‘1 , . . . ,N 3 ðtk1 Þ ¼ ‘ k1 ¼ ðq3k ðhk ðtÞÞÞm3,k1 : Continuing in this way, we obtain PðE3 Þ rðq31 3 . . . 3q3k ðhk ðtÞÞn . Since 0 o x o 1 implies q3w ðxÞ o 1, 1 r wr k, we have 0 og3 ðtÞ o 1. A.2. Proof of asymptotic equivalence (Theorem 1). We will assume that the censoring schemes are defined on the same probability space as in Section 3. We will use the following martingale inequality (Chung and Williams, 1990(1.3)): if M is a square integrable martingale and e 4 0,
ð35Þ P sup jMðsÞj 4 e r e2 EðMðtÞÞ2 : 0rsrt
In the proof of Theorem 1, we will use: Lemma 10. Let U1 ,U2 , . . . be independent uniform (0,1) random variables and let gn r n be a random sequence of positive integers, independent of the Ui sequence, such that gn -1, as n-1. If Fgn ðuÞ is the empirical distribution function of U1 ,U2 , . . . ,Ugn , then sup gn jFgn ðuÞuj ¼ OððnloglognÞ1=2 Þ,
a:s:
0rur1
¨ o+ and Re´ve´sz, 1981, Theorem 7.3.2), Gðu,nÞ, such that Proof. One can define a Kiefer process (Csorg pffiffiffiffiffi 1=2 sup j gn ðFgn ðuÞuÞgn Gðu, gn Þj-0 a:s:
ð36Þ
0rur1
¨ o+ and Re´ve´sz (1981) with aT ¼ T, we obtain as n-1. Applying Corollary 1.12.4 of Csorg lim
sup
jWðu,yÞj sup pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a:s: 1, 2TloglogT
ð37Þ
T-1 0 r y r T 0 r u r 1
where Wðu,yÞ,0 ru r 1,yZ 0, is a two-dimensional Brownian sheet. Since G has representation Gðu,yÞ ¼ Wðu,yÞuWð1,yÞ, we can replace W with G in (37). Combining this and (36), the lemma is proven. & Proof of Theorem 1. Consider the four censoring schemes constructed in Section 3. Recall that T1 ,T2 , . . . ,Tn are P independent random variables with common distribution function F. We have D ju ¼ ni¼ 1 Djui , ju ¼ 1,2,3, where Djui is defined by (19). Since each Djui of (19), is a martingale with respect to the s-field F n ðtÞ ( Corollary 6), so is D ju . By (35) " # sup jD ju ðtÞj Z e r e2 Eð½D ju ðtF Þ2 Þ,
P
ju ¼ 1,2,3:
ð38Þ
0 r t r tF
pffiffiffi To prove (11), it suffices to show that Eð½D ju ðtF Þ2 Þ r C n, for some constant C 40. From Corollary 6 n Z tF X 0 ðY0jui Y04i Þ2 dAi , Eð½D ju ðtF Þ2 Þ ¼ E/D ju ðtF Þ,D ju ðtF ÞS ¼ E i¼1
0
where A0i is defined by (14). Note that ðY0jui ðuÞY04i ðuÞÞ2 ¼ 1, if item i was censored before u by one Scheme (ju or 4) and is observed to be functioning at time u under the other Scheme (4 or ju), and = 0, otherwise. For o 2 O, the underlying probability space, let tjuiu ðoÞ ¼ inffu : jY0jui ðu, oÞY04i ðu, oÞj ¼ 1g. If Y0jui ðu, oÞ ¼ Y04i ðu, oÞ, for all u, put tjuiu ðoÞ ¼ tF . Hence, tjuiu is a discrete random variable with possible values tF and tk, 1 rk r m. Thus, Z tF Z tF 0 0 ðY0jui Y04i Þ2 dAi ¼ jY0jui Y04i j dAi rA0i ðtF ÞA0i ðtjui u Þ: tj00 i
0
We have EfA0i ðtF ÞA0i ðtjuiu Þjtjuiu g ¼ I½tjuiu o tF , a:s:, since ½tjuiu ¼ tk D ½tk oTi0 and EfA0i ðtF ÞA0i ðtk Þ jtk o Ti0 g ¼ 1. Consequently, we obtain u Þ ð39Þ Eð½D ju ðtF Þ2 Þ r Eðujun Pn u ¼ i ¼ 1 I½tjui u o tF is the number of items censored by one scheme and not the other at some time in the where ujun experiment. Put u ¼ ujun
m X k¼1
ujuk ,
ð40Þ
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
921
P ju ¼ 1,2,3, where ujuk ¼ ni¼ 1 I½tjuiu ¼ tk is the number of items observed functioning at time tk by both Schemes (ju and 4) and censored at tk by one scheme and not the other. Note that sup jY 0ju ðtÞY 04 ðtÞj rvujun ,
t r tF
ju ¼ 1,2,3
ð41Þ
Let FnðkÞ denote the empirical distribution function of fUiðkÞ : i 2 rju ðtk , oÞ \ r4 ðtk , oÞg, FnðkÞ denote the empirical distribution j0 4k j0 k function of fUiðkÞ : i 2 rju ðtk , oÞg, and let nju4k ¼ ðrju ðtk , oÞ \ r4 ðtk , oÞÞ; ju ¼ 1,2,3, 1 r kr m. Recall the notation (16). Scheme ju ¼ 1: We will prove (11) and (12) together. Recall the notation from Section 3. Note that none of the items were ðp1 Þbn11 p1 c=n11 j r n11 jFnð1Þ ðp1 Þp1 jþ 1. removed before time t1. Hence r1 ðt1 , oÞ \ r4 ðt1 , oÞ ¼ r1 ðt1 , oÞ and u11 n11 jFnð1Þ 11 11 Using Schwarz’s inequality and noting that the variance of the binomial, given n11 and p1 is n11p1(1 p1), we obtain pffiffiffi n 2 1=2 : ð42Þ Eu11 r 1 þfEn211 jFnð1Þ ðp Þp j g r 1þ 1 1 11 2 By Lemma 10, as n-1, u11 ¼ OððnloglognÞ1=2 Þg, a:s. Consider time tk. Under Schemes 1 and 4 ðkÞ ðkÞ ðkÞ ðkÞ u1k n14k jFnðkÞ ðpk ÞFnðkÞ ðUðbn Þj r n14k jFnðkÞ ðpk Þpk j þ n14k jFnðkÞ ðUðbn ÞUðbn jþ n14k jUðbn pk j 14k 14k 14k 14k 1k pk cÞ 1k pk cÞ 1k pk cÞ 1k pk cÞ
ð43Þ
ðkÞ is the bn1k pk c th-order statistic from fUiðkÞ : i 2 r1 ðtk , oÞg. Similar to (42), the expectation of the first term of where Uðbn 1k pk cÞ pffiffiffi ðkÞ ðkÞ , n jF ðkÞ ðU ðkÞ ÞUðbn j is binomial and hence (43) is bounded by n=2. For the second term, given n14k and Uðbn 1k pk cÞ pffiffiffi 1k pk cÞ 14k n14k ðbn1k pk cÞ ðkÞ , the expected value of the second term of (43) is bounded by n=2. For the third term of (43), the variance of n14k Uðbn 1k pk cÞ given n14k and n1k , is
n214k pk ð1pk Þn21k ðn1k þ 1Þ2 ðn1k þ2Þ
r npk ð1pk Þ:
Hence, again by Schwarz’s inequality, pffiffiffi ðkÞ n bn1k pk c n14k ðkÞ U þ 1, r p j r En En14k jUðbn þE k 14k ðbn p cÞ p cÞ 1k k 1k k 2 n1k þ1 n1k þ1
ð44Þ
pffiffiffi pffiffiffi Thus, Eu1k r 3 n=2þ 1, and Euu1n r ð3m2Þ=2 n þ m. Consequently, (11), for ju ¼ 1, follows from (38), (39) and (40). By Lemma 10, the first two terms on the right side of (43) equal OðnloglognÞ1=2 , a:s. For the third term, ðkÞ ðkÞ ðkÞ n14k jUðbn pk j r n14k jUðbn FnðkÞ ðUðbn Þj þ1 ¼ OððnloglognÞ1=2 Þ 1k 1k pk cÞ 1k pk ccÞ 1k pk cÞ
a:s:
ð45Þ
again by Lemma 10. Hence, for each k r m, u1k ¼ OððnloglognÞ1=2 Þ, a:s: as n-1. Consequently, (12) for ju ¼ 1, follows from this and (41). Scheme ju ¼ 2: We first prove (11). Note that, under the conditions of our Theorem, P½Y 02 ðtÞ ¼ 0,i:o: ¼ 0, tm ot o tF , by Theorem 7. This implies m2k -1, a:s:, 1 rk rm. For time t1, R1 4n21 ðp1 Þ ðp1 Þp1 j þjp1 n21 np1 Sðt1 Þj þ n21 ð1p1 ÞI½n21 o R1 þ 1: r n21 jFnð1Þ u21 n21 Fnð1Þ 21 21 n21 pffiffiffi As in the ju ¼ 1 case, the expectation of the first term of the right side of the above inequality is bounded by n þ 1. The pffiffiffi second term is p1 times the centred binomial with parameters n and S(t1) and hence its expectation is bounded by n þ 1. For the third term, P½n21 o R1 rP½nnp1 Sðt1 Þ r nn21 r P½nð1p1 ÞSðt1 Þ rðnn21 nFðt1 ÞÞ rexpf2nð1p1 Þ2 Sðt1 Þ2 g,
ð46Þ
by Hoeffding’s inequality (Petrov, 1995, p. 78). Consequently, P½n21 oR1 , i:o: ¼ 0 and the expectation of the third term is O(1). At time tk, we have ðkÞ ðkÞ ðkÞ ðkÞ u2k n24k jFnðkÞ ðpk ÞFnðkÞ ðUðR Þj rn24k jFnðkÞ ðpk Þpk jþ n24k jFnðkÞ ðUðR ÞUðR jþ n24k jUðR pk j, 24k 24k 24k 24k k 4n2k Þ k 4n2k Þ k 4n2k Þ k 4n2k Þ
ð47Þ
ðkÞ is the ðRk 4n2k Þ th order statistic from the set r2 ðtk , oÞ of n2k elements. As in the case of ju ¼ 1 above, the first where UðR k 4n2k Þ pffiffiffi two terms of the right side of (47) have expectations bounded by n þ 1. The third term is bounded by ðkÞ pk jI½n2k Z Rk : n24k ð1pk ÞI½n2k o Rk þ n24k jUðR kÞ
ð48Þ
922
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
Since ½n2k oRk D½Y 02 ðtÞ ¼ 0, where tk ot r tk þ 1 , and P½Y 02 ðtÞ ¼ 0 r fg2 ðtÞgn , 0 o g2 ðtÞ o 1, (Theorem 7), we obtain for the first term of (48): P½n2k o Rk , i:o: ¼ 0 and En24k ð1pk ÞI½n2k o Rk ¼ Oð1Þ. The second term of (48) is bounded by ðkÞ R R ðkÞ n24k jUðR pk j r n24k UðR k þ n24k k pk : ð49Þ kÞ kÞ n2k n2k pffiffiffi As with (44) and (45), the first term of (49) has expected value bounded by n=2þ 1. For the second term of (49), (see (9)) ! ! k1 k1 X X R bnrk Sðtk Þc n n n pk ¼ 24k pk n 1 r‘ Sðtk Þ 2k r n 1 r‘ Sðtk Þ 2k n24k k pk ¼ n24k n2k n2k n2k n n ‘¼1 ‘¼1 ! k 1 X n Sðtk Þ m2,k1 Sðtk Þ m2,k1 þ n 1 r‘ Sðtk Þ, r n 2k Sðtk1 Þ n m2,k1 Sðtk1 Þ n ‘¼1
where m2,k1 ¼ n2,k1 Rk1 ¼ n2,k1 bnpk1 Sðtk1 Þc. So, given n2,k 1, n2k is a binomial random variable with parameters m2,k 1 and S(tk)/S(tk 1). By Schwarz’s inequality, the first term on the right side of the above inequality has expected value pffiffiffi bounded by n=2. The second term on the right side is bounded by
n
! k2 X Sðtk Þ n2,k1 1 r‘ Sðtk1 Þ þ1: Sðtk1 Þ n ‘¼1
pffiffiffi pffiffiffi By induction, the second term has expectation bounded by n=2. Consequently, Eu2k r 6 n þ5. Hence by (38)–(40), (11) is proven (ju ¼ 2). We will prove (12) for ju ¼ 2, by showing supt jn1 Y 02 ðtÞpðtÞj ¼ Oðn1=2 ðloglognÞ1=2 Þ, a.s. directly. For 0 rt r t1 , P (no censoring has taken place), Y 02 ðtÞ ¼ ni¼ 1 I½Ti Z t. Hence by the law of the iterated logarithm for the empirical process, sup jn1 Y 02 ðtÞpðtÞj ¼ Oðn1=2 ðloglognÞ1=2 Þ
a:s:
ð50Þ
0 r t r t1
P þ For t1 o t rt2 , we can write Y 02 ðtÞ ¼ n21 m21 Em21 ðtÞ, where Em21 ðtÞ ¼ ðm21 Þ1 i2r2 ðt1 þ Þ I½t1 o Ti r t is the empirical distribution function of the m21 lifetimes Ti that are functioning and not censored at t1. These Ti have (conditional) survival function S(t)/S(t1). Then, jn1 Y 02 ðtÞpðtÞj ¼ n1 jY 02 ðtÞnð1r1 ÞSðtÞj rn1 jm21 nð1r1 ÞSðt1 Þj þn1 m21 jðEm21 ðtÞðSðt1 ÞSðtÞÞ=Sðt1 ÞÞj þ n1 jðnð1r1 ÞSðt1 Þm21 ÞðSðt1 ÞSðtÞÞ=Sðt1 Þj: Thus, sup0 r t r t2 jn1 Y 02 ðtÞpðtÞj ¼ Oðn1=2 ðloglognÞ1=2 Þ, a:s: by (50) and Lemma 10, respectively. By induction, ðloglognÞ1=2 Þ, a:s:. It is well known that sup0 r t r tF jn1 Y 04 ðtÞpðtÞj ¼ Oðn1=2
sup0 r t r tF jn1 Y 02 ðtÞpðtÞj ¼ Oðn1=2
ðloglognÞ1=2 Þ, a:s: Hence (12) is proven for ju ¼ 2. Scheme ju ¼ 3: As in the Scheme 2 case R1 4n31 ðp Þ ðp1 Þp1 j þ ð1p1 Þjn31 nSðt1 Þj þ R1 I½n31 oR1 þ 1, rn31 jFnð1Þ u31 n31 1Fnð1Þ 1 31 31 n31 pffiffiffi and a (46)-type inequality holds for P½n31 oR1 . Consequently, Eðu31 Þ ¼ Oð nÞ and u31 ¼ OððnloglognÞ1=2 Þ, a:s. At time tk, we have ðkÞ ðkÞ ðkÞ ðkÞ ðpk ÞFnðkÞ ðUðn Þj rn34k jFnðkÞ ðpk Þpk jþ n34k jFnðkÞ ðUðn ÞUðn j þ n34k jUðn pk j, u3k n34k jFnðkÞ 34k 34k 34k 34k 3k R 4n3k Þ 3k R 4n3k Þ 3k R 4n3k Þ 3k R 4n3k Þ k
k
k
k
ð51Þ ðkÞ is the ðn3k Rk 4n3k Þth order statistic from the set r3 ðtk , oÞ of n3k elements. As above, the first two terms where Uðn 3k Rk 4n3k Þ pffiffiffi of (51) are OððnloglognÞ1=2 Þ, a:s: and have expectations bounded by n þ 1. The third term of (51) is bounded by ðkÞ pk j. Similar to the ju ¼ 2 case, condition (A3) implies that rk ¼ Eðn3k Þð1pk Þ, 0 o pk o1 and Rk I½n3k o Rk þ n34k jUðn 3k Rk 4n3k Þ pffiffiffi P½n3k o Rk , i:o: ¼ 0 and EðRk I½n3k o Rk Þ ¼ Oð1Þ. Similarly, we also obtain Eu3k ¼ Oð nÞ and u3k ¼ OððnloglognÞ1=2 Þ, a:s: (11)
and (12) follows for Scheme 3.
&
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
923
A.3. Proof of Theorem 2 We first prove pffiffiffi n sup jF^ ju ðsÞF^ 4 ðsÞj-P 0,
ð52Þ
0rsrt
ju ¼ 1,2,3. We have Eq. (31), for 0 rs r t, j ¼ 1,2,3,4, where M j is defined by (6), Hj ðuÞ is F nðjÞ ðuÞ-predictable and pffiffiffi nsup0 r s r t jBj ðsÞj ¼ oP ð1Þ, as n-1, by (32) and Theorem 7 (Fleming and Harrington, 1991, Proof of Theorem 6.3.1). Hence for ju ¼ 1,2,3, Z s Z s pffiffiffi pffiffiffi pffiffiffi njF^ ju ðsÞF^ 4 ðsÞj rS0 ðsÞ Hju ðuÞd nM ju ðuÞ H4 ðuÞd nM 4 ðuÞ þ oP ð1Þ: 0
0
By (35) and Corollary 5, for j ¼ 1,2,3,4, Z s
Z t pffiffiffi Hj ðuÞðnpðuÞÞ1 d nM j ðuÞ 4 e r En1 e2 ðnHj ðuÞðpðuÞÞ1 Þ2 dA j ðuÞ: P sup 0rsrt
0
ð53Þ
0
Using (12) and the fact that supu r t jn1 Y 04 ðuÞpðuÞj-P 0, we obtain sup0 r u r t jn1 Y 0ju ðuÞpðuÞj-P 0, ju ¼ 1,2,3 and Z t ðnHj ðuÞðpðuÞÞ1 Þ2 dA j ðuÞ r n1 A j ðtÞ sup ðnHj ðuÞðpðuÞÞ1 Þ2 ¼ oð1Þ a:s: n1 e2 0
0rurt
Since the above is uniformly bounded (in n) (Billingsley, 1995, Theorem 16.5), (53) converges to 0 as n-1. Finally, using the integration by parts formula (Fleming and Harrington, 1991, Theorem A.1.2), Z s pffiffiffi 1 1 2 sup d nðM ju ðuÞM 4 ðuÞÞ r pffiffiffi sup jD ju ðsÞj 1 : pðtÞ n0 r s r t 0 r s r t 0 npðuÞ By Theorem 1, the above supremum converges to 0, in probability and hence (52) is proven. Consider 2 nI½Y ðuÞ 40 sup0 r u r t jS^ j0 ðuÞS20 ðuÞj 1 1 0ju sup jðHju ðuÞÞ2 Y 0ju ðuÞðpðuÞÞ1 j r sup þ : S0 ðtÞ 0 r u r t pðuÞ S20 ðtÞpðtÞ Y 0ju ðuÞ 0rurt
ð54Þ
By (12), (29), (30) and the fact that n1 sup0 r u r t jY 04 ðuÞpðuÞj-P 0, we obtain that the left side of (54) converges to 0, in Rs probability. It follows that 0 ðHju ðuÞÞ2 Y 0ju ðuÞdL0 ðuÞ-P vðsÞ, and Theorem 2 is proven. & References Andersen, P.K., Borgan, Ø., Gill, R.D., Keiding, N., 1993. Statistical Models Based on Counting Processes. Springer-Verlag, New York. Bagdonavicˇius, V., Nikulin, M., 2002. Accelerated Life Models: Modeling and Statistical Analysis. Chapman & Hall/CRC, Boca Raton, Fl. ¨ Balakrishnan, N., Aggarwala, R., 2000. Progressive Censoring: Theory, Methods, and Applications. Birkhauser, Boston. Billingsley, P., 1995. Probability and Measure, third ed. Wiley, New York. Bordes, L., 2004. Non-parametric estimation under progressive censoring. J. Statist. Plann. Inference 119, 171–189. + S., Horva´th, L., 1981. Strong approximations of some biometric estimates under random censorship. Z. Wahrsch. verw. Geb. 56, ¨ o, Burke, M.D., Csorg 87–112. ¨ Chung, K.L., Williams, R.J., 1990. Introduction to Stochastic Integration, second ed. Birkhauser, Boston. + S., Horva´th, L., 1986. Confidence bands for censored samples. Canad. J. Statist. 14, 131–144. ¨ o, Csorg + M., Re´ve´sz, P., 1981. Strong Approximations in Probability and Statistics. Academic Press, New York. ¨ o, Csorg Fleming, T.R., Harrington, D.P., 1991. Counting Processes and Survival Analysis. Wiley, New York. Gill, R.D., 1980. Censoring and stochastic integrals, Mathematical Centre Tracts, vol. 124, Mathematisch Centrum, Amsterdam. Gillespie, M., Fisher, L., 1979. Confidence bands for the Kaplan–Meier survival curve estimates. Ann. Statist. 7, 920–924. Hall, W.J., Wellner, J.A., 1980. Confidence bands for a survival curve from censored data. Biometrika 67, 133–143. Petrov, V.V., 1995. Limit Theorems of Probability Theory. Oxford University Press, Oxford.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Nonparametric estimation of a survival function under progressive Type-I multistage censoring$ M.D. Burke Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada T2N 1N4
a r t i c l e i n f o
abstract
Article history: Received 19 May 2009 Received in revised form 12 August 2010 Accepted 19 August 2010 Available online 24 August 2010
Failure time data subject to three progressive Type-I multistage censoring schemes are studied. Product limit estimators are proposed for the estimation of the survival function. It is shown that the resulting estimators are asymptotically equivalent to the corresponding estimators where the data are subject to a random iid right censoring scheme. Many well-known results on confidence bands and tests for randomly right censored data hold for these progressive censoring schemes. & 2010 Elsevier B.V. All rights reserved.
Keywords: Progressive multistage censoring Martingales Product limit estimation Asymptotic distribution
1. Introduction Experimental models using progressive censoring can be used in reliability analysis, product testing and clinical trials. In many situations, the precise estimation of the left tail of the distribution of failure time is of great importance. Hence, one would wish, at the beginning of the study, to test a large number of items (patients). In the reliability setting, to reduce the total time on test, the experimenter, at specified times, purposely and randomly removes from the test items that are still functioning. This is called progressive Type-I multistage censoring and it is the subject of this paper. One reason to purposely remove items is that they are costly to produce. A large proportion of them can be removed at a predetermined time. The items so removed can then be sold and re-used. Another reason to purposely remove functioning items is that the operation of the test may be costly. A third reason is that one may need to free up some laboratory space for other purposes. The removal (censoring) times can be chosen to correspond to important times such as various warranty periods. At these and other removal times, the removed items can be disassembled and examined for wear using destructive testing methods. We refer to Balakrishnan and Aggarwala (2000) for further applications in reliability. In clinical trials, restricting patient loads for physicians may be a factor in keeping costs down. Also, physicians may desire to spend more resources on new patients than on patients who have survived past certain benchmarks. Thus, removing all but a certain number of patients from the study at certain times is desirable. In this paper, we will develop an asymptotic theory for three types of progressive censored data. It is shown that martingale theory can be applied to these progressively censored data (Section 3). In addition, it is shown that the martingale processes for such data are asymptotically equivalent to a martingale process for data which are randomly right censored by independent and identically distributed random variables with a particular distribution function (Theorem 1).
$
Research partially supported by a Canadian NSERC Discovery Grant.
Tel.: +1 403 220 3959.
E-mail address: [email protected] 0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.08.009
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
911
Such randomly right censored data have been studied extensively (Fleming and Harrington, 1991; Andersen et al., 1993). Using this asymptotic equivalence, many well-known results on confidence bands and tests for randomly right censored data are derived for these progressive censoring schemes. The results are applied to the nonparametric estimation of the survival function under progressive censorship and, in particular, the product limit estimator. Suppose there are n items that are put in use at time t ¼ 0. For item i, let T0i be the time to failure. We will consider three progressive censoring schemes and a corresponding random right censoring scheme. Let t1 o t2 o otm be fixed times when some items will be removed from study (censored). Assume that the T0i are independent random variables with a common continuous distribution function F0, cumulative hazard function L0 , and survival function S0 ¼ 1F0 . Let tF0 ¼ infft : F0 ðtÞ ¼ 1g. We will also assume that the T0i are, at first, subject to random right censorship by an independent sequence C0i of iid random variables with distribution function G0. If no progressive censoring occurs, one observes Ti ¼ Ti0 4C0i and d0i ¼ I½Ti0 r C0i . The survival function of the Ti is SðtÞ ¼ P½Ti 4t ¼ ð1G0 ðtÞÞS0 ðtÞ:
ð1Þ
The Ti have support ½0, tF , where tF ¼ infft : FðtÞ ¼ 1g, F ¼ 1S. Let bxc denote the greatest integer less than or equal to x and dxe denote the smallest integer greater than or equal to x. The ðTi , d0i Þ are subject to progressive censoring. The three such progressive schemes are: Scheme 1: (Progressive Type-p). For the n‘ items still observed to be functioning at time t‘ , randomly select, independent of the future values of Ti, bn‘ p‘ c of them to be removed (censored), where p‘ is a predetermined proportion, 0 r p‘ o1. Scheme 2: (Progressive Type-R). For the n‘ items still observed to be functioning at time t‘ , randomly select, independent of the future values of Ti, R‘ of them to be removed (censored). If fewer than R‘ items are on trial at t‘ , remove all the remaining ones. Scheme 3: (Progressive Type-R*). For the n‘ items still observed to be functioning at time t‘ , randomly select, independent of the future values of Ti, n‘ ðR‘ 4n‘ Þ of them to be removed (censored). (If n‘ rR‘ , no items are removed.) In Schemes 1 and 2 the proportion p‘ or the number R‘ of items to be removed are specified. In Scheme 3, one specifies the number R‘ to remain in the study at time t‘ . This is useful when one wants to ensure that there are sufficient items on trial for the asymptotics to hold. For Schemes 2 and 3, R‘ and R‘ are predetermined fixed numbers. If item i is removed at time t‘ under Scheme j, j ¼ 1,2,3, then let Cji ¼ t‘ . If Ti is observed, then we will put Cji ¼ tF , so that events of the form ½t o Cji ¼ ½Cji % / t, ½t rCji ¼ ½Cji 5t and ½Ti r Cji ¼ ½Cji 5Ti are well defined. One observes Xji ¼ minfTi , Cji g, dji ¼ I½Xji ¼ Ti and dji d0i ¼ I½Xji ¼ Ti0 , i ¼ 1,2, . . . ,n. The schemes above are Type-I censoring. However, they are different from the Type-I scheme in Bagdonavicˇius and Nikulin (2002, p. 79) where the censoring variables are predetermined non-random values and a special case of right censoring by independent random variables. The book by Balakrishnan and Aggarwala (2000) studies order statistics and parametric inference for progressive Type-II multistage censoring, where the times of censoring are order statistics of the lifetimes and the numbers removed are specified in advance. As pointed out by them (see p. 10), Type-I, and not Type-II censoring, is usually carried out in practice. Bordes (2004) established large sample properties of the product limit estimator under progressive Type-II censoring. The nature of the processes and their limit distributions are different in the Type-I and Type-II cases. In order to develop a proper asymptotic result for the Type-R and Type-R* models, we will assume that the numbers R‘ and R‘ are proportional to n, the sample size. (If R‘ is fixed and n tends to infinity, the effect of censoring is asymptotically negligible.) Note that if too many items are removed, the remaining items may be too few in number for the asymptotic theory to hold after time t‘ . In this case, inference about F0 would be valid up to time t‘ 4tF0 , only. We will assume that the R‘ of Scheme 2 satisfy R‘ ¼ bnr ‘uc,
‘ ¼ 1,2, . . . ,m,
ð2Þ
where the r‘u Z 0 are predetermined. For Scheme 3, we will assume R‘ ¼ dnr ‘ e, ‘ ¼ 1,2, . . . ,m
and
0 o r‘ or‘1 o1, ‘ ¼ 2, . . . ,m:
ð3Þ
Note that if r‘1 r r‘ , then no censoring will take place at t‘ . Let G denote the discrete (sub)-distribution function with probability mass function
dGðt1 Þ ¼ p1 ,
dGðt‘ Þ ¼ p‘
‘1 Y iu ¼ 1
ð1piu Þ,
dGðtF Þ ¼ 1
m X
dGðt‘ Þ,
ð4Þ
‘¼1
‘ ¼ 2, . . . ,m. If tF ¼ 1 and dGðtF Þ 4 0, then G is a sub-distribution function. Define the function p by
pðtÞ ¼ SðtÞð1GðtÞÞ, t Z0,
ð5Þ
where S is defined by (1). One can define an iid random censoring scheme. Scheme 4: (IID random). Let C4i be a sequence of independent and identically distributed random variables having distribution function G. One observes X4i ¼ minfTi ,C4i g, d4i ¼ I½X4i ¼ Ti and d4i d0i ¼ I½X4i ¼ Ti0 , i ¼ 1,2, . . . ,n.
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Using counting process notation, for j ¼ 1,2,3,4 and i ¼ 1,2, . . . ,n, let N0ji ðtÞ ¼ I½Ti0 r t,Ti0 r C0i 4Cji ;
N 0j ¼
n X
N0ji ;
i¼1
Y0ji ðtÞ ¼ I½t r Ti0 4C0i 4Cji ;
n X
Y 0j ¼
Y0ji ;
i¼1
A0ji ðtÞ ¼
Z 0
t
Y0ji ðuÞ dL0 ðuÞ;
Aj ¼
n X
A0ji ;
i¼1
M0ji ðtÞ ¼ N0ji ðtÞA0ji ðtÞ;
Mj ¼
n X
M0ji :
ð6Þ
i¼1
When censoring Scheme j is used, N0ji has a jump of size 1 at the observed failure time T0i of item i and Y0ji(t) indicates whether item i is in use (on test) at time t. In the following, we will assume one of the three schemes is used for the whole experiment. However, our results will hold if one combines the different schemes for the same experiment, for example, remove all but R*1 at t1 (Scheme 3) and then remove R2 items at t2 (Scheme 2). The organization of this article is as follows: We state the main results in Section 2. In Section 3, we define Schemes 1–4 explicitly on the same probability space and prove a number of martingale properties that are used in our proofs of the main results. In Section 4, expressions for the bias of our estimators are obtained (Theorem 7) and consistency of our estimators is established (Theorem 8). A small simulation is conducted in Section 5 to demonstrate the applicability of the theory. Proofs of results on bias, as well as Theorems 1 and 2, are given in the Appendix (Section Appendix A). 2. Main results The main result of this paper is Theorem 1 which states that the censoring schemes can be defined on the Rt same probability space such that M j ¼ N 0j 0 Y 0j ðuÞdL0 ðuÞ, j = 1,2,3,4 satisfy supt jMj ðtÞM4u ðtÞj ¼ OP ðn1=4 Þ. As a consequence, one can deduce that the product limit estimators under the different schemes have the same asymptotic distribution (Theorem 2). We define the product limit estimators ( ) Y DN 0j ðuÞ ^S j ðtÞ ¼ 1 , j ¼ 1,2,3,4, ð7Þ Y 0j ðuÞ urt where DN 0j ðuÞ ¼ N 0j ðuÞN 0j ðuÞ is the number of T0i that are observed to occur at time u. Since the failure time distribution is continuous, this number is either 0 or 1. We also define Nelson–Aalen estimators, j= 1,2,3,4, for the common L0 ðtÞ by Z t n X dN 0j ðuÞ ^ ðtÞ ¼ L I½Ti0 rt,Ti0 rC0i 4Cji ðY 0j ðTi0 ÞÞ1 : ð8Þ ¼ j Y ðuÞ 0 0j i¼1 We will assume, according to which scheme is used: (A1) 0 rp1 ,p2 , . . . ,pm o1. P (A2) r m m j ¼ 1 r‘ o 1, where r‘ ¼ r‘u=Sðt‘ Þ,‘ ¼ 1, . . . ,m. (A3) 0 or1 r Sðt1 Þ; 0 or‘ =r‘1 r Sðt‘ Þ=Sðt‘1 Þ,‘ ¼ 2, . . . ,m: Note that assumptions (A2) and (A3) depend on the unknown S, while (A1) does not. Asymptotic equivalence of Schemes 1 and 2 (Theorem 1) will hold when p‘ ¼ r‘ ð1r ‘1 Þ1 , 2 r‘ rm, ð9Þ Q‘1 or equivalently, when r‘ ¼ p‘ j ¼ 1 ð1pj Þ. Condition (9) equates the expected proportions of items removed for Schemes 1 and 2. For Scheme 3, we will let (equating expected proportions) p1 ¼ r1 ,
ð1p1 Þ ¼
r1 ; Sðt1 Þ
ð1p‘ Þ ¼
r‘ Sðt‘1 Þ Sðt Þ , ‘ ¼ 2, . . . ,m, r‘1 ‘
ð10Þ
Q that is, when r‘ ¼ Sðt‘ Þ ‘k ¼ 1 ð1pk Þ. If (9) and (10) hold, assumptions (A1)–(A3) are equivalent. The following theorem shows that the three schemes are asymptotically equivalent. Recall the notation given by (6). Theorem 1. Suppose that (Aj) holds for censoring Scheme j , j ¼ 1,2,3. One can define a probability space, on which T0i , Cji, C4i, i ¼ 1,2, . . . ,n, and filtration F n ðtÞ are defined, such that C4i are iid with distribution function (4), D j ðtÞ ¼ M j ðtÞM 4 ðtÞ is an
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
913
F n ðtÞ-martingale and sup jD j ðtÞj ¼ OP ðn1=4 Þ,
ð11Þ
0 r t r tF
Y ðtÞ Y ðtÞ 0j 04 sup ¼ a:s: Oðn1=2 ðloglognÞ1=2 Þ: n 0 r t r tF n
ð12Þ
Under (A2), p‘ is defined by (9). Under (A3), p‘ is defined by (10). The construction and martingale properties in the statement of Theorem 1 are detailed in Section 3. The convergence statements (11) and (12) are proven in the Appendix (Section Appendix A). As a consequence of Theorem 1, we obtain the following weak convergence result for progressively censored data. Theorem 2. Let T10 ,T20 , . . . ,Tn0 be independent random variables each with continuous distribution function F0 and subject to iid censoring by C01 , . . . ,C0n , and progressive censoring under Scheme j, j ¼ 1,2,3. Suppose assumption (Aj) holds for Scheme j. Then, for any t satisfying SðtÞ 4 0, pffiffiffi nðF^ j F0 Þ-D ð1F0 ÞWðvÞ, Rs j ¼ 1,2,3, in the space D[0,t], as n-1, where W is a Wiener process and vðsÞ ¼ 0 ðpðuÞÞ1 dL0 ðuÞ, with p defined by (5) and the p‘ , given by (9) for j =2, resp. (10) for j =3. Also, for j =1,2,3, pffiffiffi ^ nðF j F0 Þ -D WðvÞ: ð1F^ j Þ The proof of Theorem 2 is given in the Appendix (Section Appendix A). While the experimenter knows the values of pk, Rk and R*k (1 rk r m), the distribution functions F0 and G0, L0 and v are all unknown. Uniform confidence bands can be obtained directly using Theorem 2 and Z s ððY 0j ðuÞ1ÞY 0j ðuÞÞ1 dN 0j ðuÞ, j ¼ 1,2,3, v^ j ðsÞ ¼ n 0
to estimate v. In the iid random censorship case, such bands were given by Gill (1980). See also the related bands (Gillespie ¨ o+ and Horva´th, 1986; Burke et al., 1981, p. 107). One can also obtain the following Hall–Wellnerand Fisher, 1979; Csorg type bands (Hall and Wellner, 1980), which were shown to have good moderate sample size properties. Corollary 3. Under the conditions of Theorem 2, n1=2 ðF^ j F0 Þ ð1 þ v^ j Þð1F^ j Þ
-D W o ðK0 Þ,
j =1,2,3, as n-1, on D[0,t], where K0 ðsÞ ¼ vðsÞ=ð1 þvðsÞÞ and Wo is a Brownian bridge process on [0,1]. Hence, for a ¼ K0 ðtÞ, o ^ j ðsÞÞ1 ðS^ j ðsÞÞÞ1 S^ j ðsÞ 7n1=2 cð1 aÞ ðð1 þ v
ð13Þ
o are uniform (s rt) confidence bands for S0 ¼ 1F0 , where j ¼ 1,2,3, where P½sup0 r x r a jW o ðxÞj r cð1 aÞ ¼ 1a.
A simulation is conducted in Section 5 using (13). One can also obtain confidence bands for the cumulative hazard function, based on the Nelson–Aalen estimator (Andersen et al., 1993, Sections 4.1 and 4.2). 3. Construction and some martingale results Let Ni0 ðtÞ ¼ I½Ti0 rt,
A0i ðtÞ ¼
Z
t 0
I½u r Ti0 dL0 ðuÞ,
Mi0 ¼ Ni0 A0i ,
ð14Þ
i= 1,2,y,n. Although not observable under the censoring schemes, these processes are the elementary objects that are used to prove the martingale properties of the censored processes. It is well known that M0i is a martingale with respect to its natural filtration sfI½Ti0 r u : u rtg and with respect to F ðnÞ ðtÞ ¼ sfI½Ti0 r u : u rt, i ¼ 1,2, . . . ,ng:
ð15Þ F n ðtÞ
of (17), which contains It is shown (Theorem 4) that it is also a martingale with respect to the enlarged filtration F ðnÞ ðtÞ, the censoring information, up to time t, by the C0i, the three progressive censoring schemes and a random right censoring scheme using iid random variables all defined below. Some consequences are that the corresponding processes M0ji defined by (6) for each of the four censoring schemes are martingales (Corollary 5). Moreover, the differences, M0jui M04i , ju ¼ 1,2,3 are martingales with respect to F n ðtÞ ( Corollary 6). This is used in Appendix A to prove asymptotic equivalence of the schemes (proof of Theorem 1).
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We will define the three progressive censoring schemes explicitly as follows: Let fUiðkÞ : k ¼ 1,2, . . . ,m; i ¼ 1,2, . . . ,ng be a rectangular array of independent uniform (0,1) random variables, independent of the Ti ¼ Ti0 4C0i , i ¼ 1,2, . . . ,n. We will define (risk) sets of indices rj ðtk , oÞ and censoring variables Cji ðoÞ, recursively, for each Scheme j, time tk, and item i, where j ¼ 1,2,3; k ¼ 1,2, . . . ,m; i ¼ 1,2, . . . ,n. Put rj ðt1 , oÞ ¼ fi : t1 rTi ðoÞg for each j ¼ 1,2,3. Notation: njk #rj ðtk , oÞ ¼ Y 0j ðtk Þ;
mjk #rj ðtk þ, oÞ ¼ Y 0j ðtk þÞ:
ð16Þ ð1Þ Uið1Þ r Uðbn , 11 p1 cÞ
ð1Þ Uðbn 11 p1 cÞ
where is Scheme j =1: (Progressive Type-p). Item with index i 2 r1 ðt1 , oÞ is removed at time t1, if the bn11 p1 c-order statistic from the set fUið1Þ : i 2 r1 ðt1 , oÞg, and put C1i ¼ t1 , for this i. Let r1 ðt2 , oÞ ¼ fi : t2 rTi ðoÞ,C1i ðoÞat1 g denote the risk set of the items at time t2. ðkÞ We continue in this way for k ¼ 2, . . . ,m, where item with index i 2 r1 ðtk , oÞ is removed at time tk, if UiðkÞ rUðbn , where 1k pk cÞ ðkÞ Uðbn 1k pk cÞ
is the bn1k pk c-order statistic from the set fUiðkÞ : i 2 r1 ðtk , oÞg, and put C1i ¼ tk . We define r1 ðtk þ 1 , oÞ ¼ fi : tk þ 1 r
Ti ðoÞ,C1i ðoÞ tk g, for k rm1. Finally, put C1i ðoÞ ¼ tF , if item i has not been removed at times t1 , . . . ,tm under Scheme 1. Scheme j =2: (Progressive Type-R). For k ¼ 1,2 . . . ,m, recursively, item with index i 2 r2 ðtk , oÞ is removed at time tk, if ðkÞ ðkÞ UiðkÞ r UðR , where UðR is the Rk-th order statistic from the set fUiðkÞ : i 2 r2 ðtk , oÞg. Put C2i ¼ tk . We define r2 ðtk þ 1 , oÞ ¼ kÞ kÞ
fi : tk þ 1 r Ti ðoÞ, C2i ðoÞ tk g, for kr m1. Put C2i ðoÞ ¼ tF , if item i has not been removed at times t1 , . . . ,tm under Scheme 2. ðkÞ Scheme j ¼ 3: (Progressive Type-R*). For k ¼ 1,2 . . . ,m, recursively, item i 2 r3 ðtk , oÞ is removed at time tk, if UiðkÞ r Uððn
þ jk Rk Þ
where
ðkÞ Uððn þ jk Rk Þ Þ
is the ðnjk Rk Þ þ th order statistic from the set
fUiðkÞ
Þ
,
: i 2 r3 ðtk , oÞg. Put C3i ¼ tk . Define r3 ðtk þ 1 , oÞ ¼
fi : tk þ 1 r Ti ðoÞ, C3i ðoÞ tk g, for k rm1. Put C3i ðoÞ ¼ tF , if item i has not been removed at times t1 , . . . ,tm under Scheme 3. Then, N0ji ðtÞ ¼ I½Ti0 r t,Ti0 r C0i 4Cji and Y0ji ðtÞ ¼ I½Ti 4Cji Zt, i ¼ 1,2, . . . ,n, have the required joint distribution for Scheme j, j ¼ 1,2,3. For these types of censoring, it is important to note two things: Firstly, events of the form ½Cji ¼ tk ,tk oTi depend on the risk set rj ðtk , oÞ, but do not depend on the future values of the Tiu , namely Tiu I½Tiu 4tk , iu ¼ 1,2, . . . ,n. 0
That is, ½Cji ¼ tk ,tk oTi 2 sfI½Tiu rtku ,Uiðk0 Þ ,1 r ku r k,1 r iu r ng: Secondly, for tk os, P½s o Ti0 jtk oTi ,Cji ¼ tk ¼ P½s oTi0 jtk oTi ¼ P½s o Ti0
jtk o Ti0 ,
that is, knowing that item i would be censored at tk, under Scheme j, does not affect the conditional failure time distribution of T0i , nor does it change the conditional failure time distribution of those functioning items that are not removed. We will compare these progressive censoring schemes with the iid scheme. Scheme 4 (IID random) For the same array U(k) i , 1 r kr m, i ¼ 1,2, . . . ,n, define the random variables C4i as 8 ð1Þ > t1 if Ui r p1 , > > < ðkÞ ð‘Þ C4i ¼ t‘ if pk o Ui for k o‘ and Ui r p‘ , > > ð‘Þ > :t if p oU for all ‘ r m: F
‘
i
The C4i have distribution (4) and are independent of the failure times Ti. Let F 3G denote the smallest s-field, containing the sets in F and G. With the Cji as defined in this Section and using the notation (6), define, for i ¼ 1,2, . . . ,n and j ¼ 1, 2, 3, 4, F ji ðtÞ ¼ sfN0ji ðuÞ,I½Ti 4Cji r u,Cji o Ti , I½Ti 4Cji ru,C0i oTi0 4Cji : 0 ru rtg, and F nðjÞ ðtÞ ¼
Wn
i¼1
F ji ðtÞ, the smallest s-field containing all the events in F ji ðtÞ, 1r i rn. The filtration F ðjÞ n ðtÞ is right
continuous and M j ðtÞ is adapted to it. In order to compare the processes under different censoring schemes, we will consider the larger filtration: 4 _ ðF nðjÞ ðtÞÞ, ð17Þ F n ðtÞ ¼ F ðnÞ ðtÞ3 j¼1
where F ðnÞ ðtÞ is defined by (15). Eq. (17) contains all the information concerning which items are censored up to time t as well as the failure times of the items, whether censored by one or more of the schemes or not, up to time t. Consider the following thought experiment. Instead of removing a functioning item at time tk when it is censored, place a card beside it indicating the time tk and the scheme (j = 0, 1, 2, 3 or 4) under which it is censored. Each item may have one or more (up to five) cards placed beside it. In this way, its lifetime after censoring can still be observed. The filtration F n ðtÞ contains all this information regarding failure times, up to time t, and which items are carded and by which scheme, up to time t. These remarks and the following theorem show that Schemes 1–3 satisfy the general definition of independent censoring (Gill, 1980; Andersen et al., 1993, p. 139). Theorem 4. The processes M0i of (14) and variation process /Mi0 ,Mi0 SðtÞ ¼ A0i ðtÞ;
Pn
i¼1
Mi0 are martingales with respect to the filtration F n ðtÞ. M0i has quadratic
/Mi0 ,Mi00 SðtÞ ¼ 0, iaiu:
ð18Þ
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
915
Corollary 5. The processes M0ji(t) of (6) are martingales with respect to F n ðtÞ and hence with respect to the filtration F ðjÞ n ðtÞ with quadratic variation process /M0ji ,M0ji SðtÞ ¼ A0ji ðtÞ;
/M0ji ,M0jiu SðtÞ ¼ 0, iaiu,
where A0ji(t) is defined by (6). Moreover, M j is an F n ðtÞ-martingale with quadratic variation process /M j ,M j SðtÞ ¼ A j ðtÞ. Corollary 6. The processes Djui ðtÞ ¼ M0jui ðtÞM04i ðtÞ, are zero-mean
ju ¼ 1,2,3, i ¼ 1,2, . . . ,n,
F n ðtÞ-martingales
/Djui ,Djui SðtÞ ¼
Z
t
ð19Þ
with quadratic variation process
ðY0jui Y04i Þ2 dL0 ;
/Djui ,Djuiu SðtÞ ¼ 0 for iaiu:
0
Moreover, D ju
Pn
i¼1
Djui is an F n ðtÞ-martingale having quadratic variation process /D ju ,D ju SðtÞ ¼
Pn
i¼1
Rt
2 0 ðY0jui Y04i Þ dL0 .
Proof of Theorem 4. Clearly, M0i (t) is adapted to the filtration F n ðtÞ. The sfield F n ðtÞ is generated by sets of the form T T G ¼ ni0 ¼ 1 ð 4j ¼ 1 Bjiu \ B0jiu \ Jiu Þ, where Jiu ¼ ½Ti00 ru, for some u r t, or ½t oTi00 , B0jiu ¼ ½Tiu 4Cjiu r u,C0iu oTi00 4Cjiu , for some u rt, or their complements, and Bjiu ¼ ½Tiu 4Cjiu r u, Cjiu o Tiu , for some u rt, or their complements. We split the following integral into two parts Z G
Mi0 ðtuÞMi0 ðtÞ dP ¼
Z
Z þ
G\½Ti0 r t
G\½t o Ti0
fMi0 ðtuÞMi0 ðtÞg dP:
ð20Þ
The first term on the right side of (20) equals 0, since IG\½T 0 r t ðMi0 ðtuÞMi0 ðtÞÞ ¼ 0, a:s: For the second term of (20), let i T T Giu ¼ iuai ð 4j ¼ 1 Bjiu \ B0jiu \ Jiu Þ \ Bji \ B0ji . Then, Z Z Z Ni0 ðtuÞNi0 ðtÞ dP ¼ IGui I½t oTi0 fNi0 ðtuÞNi0 ðtÞg dP ¼ PðGuÞ I½t oTi0 fNi0 ðtuÞNi0 ðtÞg dP ¼ PðGiuÞ½F0 ðtuÞF0 ðtÞ: ð21Þ i G\½t o Ti0
The second equality follows because T0i and the Ti00 are independent and the five censoring mechanisms, up to time t, do not depend on the value of T0i when ½t o Ti0 . Note also that ½Ti 4Cji ru,Cji o Ti , for some u rt \ ½t oTi0 ¼ ½Cji r t,Cji o C0i \ ½t o Ti0 and ½Ti 4Cji r u,C0i oTi0 4Cji ,for someu r t \ ½t o Ti0 ¼ ½C0i rt,C0i o Cji \ ½t o Ti0 . We also have Z G\½t o Ti0
Z
t0 t
I½u r Ti0 dL0 ðuÞ dP ¼
Z
IGiu
¼ PðGiuÞ
Z
t0
t
Z
I½t oTi0 I½u rTi0 dL0 ðuÞ dP ¼
Z
IGiu
Z t
t0
I½u r Ti0 dL0 ðuÞ dP
t0
ð1F0 ðuÞÞ dL0 ðuÞ dP, t
which equals the right side of (21). Hence the second term of (20) equals zero. Since the sets G form a p-system which generates F n ðtÞ, E½Mi0 ðtuÞMi0 ðtÞjF n ðtÞ ¼ 0, almost surely, (Billingsley, 1995, Theorem 34.1). Consequently, Pn M0i (t) is a martingale. The sum of martingales, i ¼ 1 Mi ðtÞ, with respect to the same filtration, is clearly a martingale. Consider the vector ðN1 ,N2 , . . . ,Nn Þ. Since each Ni is a counting process and the jumps of any two of them cannot occur simultaneously, with probability 1, this vector is a multivariate counting process. Hence (Gill, 1980, Appendix 1) Z t Z t ð1DAi ÞdAi ; /Mi ,Miu SðtÞ ¼ DAi dAiu , iaiu /Mi ,Mi SðtÞ ¼ 0
0
Since the distribution F0 is assumed to be continuous, (18) holds and hence the theorem.
&
Proof of Corollary 5. Since the process Y0ji(t) is measurable with respect to F nðjÞ ðtÞ D F n ðtÞ and left continuous, it is Rt 0 predictable for both s-fields. Consequently, M0ji ðtÞ ¼ 0 Y0ji ðuÞ dM i ðuÞ are F n ðtÞ-martingales. Moreover, since M0ji(t) is ðjÞ F ðjÞ n ðtÞ-measurable, each M0ji(t) is an F n ðtÞ-martingale. From stochastic integration theory Z Z Z Y0ji Y0jiu d Mi ,Miu Y0ji ðuÞ dMi ðuÞ, Y0jiu ðuÞ dMiu ðuÞ ¼
and the second statement of Corollary 5 follows from (18).
&
The proof of Corollary 6 is similar, since Y0jui ðtÞY04i ðtÞ is left continuous and F n ðtÞ-measurable implies that Rt 0 Djui ðtÞ ¼ 0 ðY0jui Y04i Þ dMi is an F n ðtÞ-martingale.
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M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
^ 4. Bias and consistency of S^ j and L j Having established that M0ji and M j of (6) are martingales, we can use standard martingale arguments to derive properties of our estimators. Theorem 7 concerns a bound on the bias of S^ j , j ¼ 1,2,3. Let t0 ¼ 0, tm þ 1 ¼ tF and p0 ¼ 0. Define, for k ¼ 1,2, . . . ,m, hk ðuÞ ¼ ðSðtk ÞSðuÞÞ=Sðtk Þ,
u Z0,
q1k ðxÞ ¼ hk1 ðtk Þ þ xð1pk Þ Sðtk Þ=Sðtk1 Þ,
q3k ðxÞ ¼ hk1 ðtk Þ þ xrk Sðtk Þ=Sðtk1 Þ,
x Z0,
x Z 0:
ð22Þ
Theorem 7. If SðtÞ 40 and (A1) holds for Scheme 1, (resp. (A2) holds for Scheme 2 and (A3) holds for Scheme 3), then 0 r ES^ j ðtÞS0 ðtÞ r f1S0 ðtÞgP½Y 0j ðtÞ ¼ 0 r f1S0 ðtÞgfgj ðtÞgn ,
ð24Þ
and gj ðtÞ o 1, where gj is defined by ( 1SðtÞ g1 ðtÞ ¼ q11 3q12 3 3q1k 3hk ðtÞ
g2 ðtÞ ¼
k Y
FðnðtÞÞ
g3 ðtÞ ¼
if t rt1 if tk o t rtk þ 1 ;
Sðtw Þ )rw ! Sðtw Þ o1; FðnðtÞÞFðtw Þ
ð25Þ
(
w¼1
(
ð23Þ
1SðtÞ
if t rt1
q31 3q32 3 3q3k 3hk ðtÞ
if tk o t rtk þ 1 ;
ð26Þ
ð27Þ
Hence, S^ j ðtÞ is an asymptotically unbiased estimator of S(t) with bias converging to zero at an exponential rate as n-1. Theorem 7 is well known when iid censoring is used (Fleming and Harrington, 1991, Lemma 3.2.1). In this case, P½Y 04 ðuÞ ¼ 0 ¼ fpðuÞgn , where p is defined by (5). The proof of Theorem 7 is in the Appendix. Consider I½Y 0j ðtÞ 40=Y 0j ðtÞ, which is taken to be zero if the numerator is zero, j =1,2,3,4. Because of left continuity, I½Y 0j ðtÞ 40=Y 0j ðtÞ is predictable with respect to F ðjÞ n ðtÞ. Consequently, for the Nelson–Aalen estimators (8), on letting R Lj ðtÞ ¼ 0t I½Y 0j ðuÞ 4 0 dL0 ðuÞ, ^ ðtÞL ðtÞ ¼ L j j
Z
t 0
I½Y 0j ðuÞ 4 0ðY 0j ðuÞÞ1 dM j ðuÞ
^ ^ is a well-defined stochastic integral and a F ðjÞ n ðtÞmartingale. Hence, EfL j ðtÞLj ðtÞg ¼ 0, and the bias EfL j ðtÞLðtÞg ¼ Rt ^ ðtÞ is an asymptotically unbiased estimator of 0 P½Y 0j ðuÞ ¼ 0dL0 ðuÞ. Using (24), it follows that for Schemes ju ¼ 1,2,3, L ju LðtÞ, with bias converging to zero at an exponential rate, as n-1, for t 2 ½0, tF Þ, under the conditions of Theorem 7. The following establishes weak consistency for the Nelson–Aalen and product limit estimators under progressive censoring Schemes 1–3.
Theorem 8. (a) If t 2 ð0, tF satisfies Y 0j ðtÞ-P 1
as n-1,
ð28Þ
then ^ ðsÞL0 ðsÞj-P 0, sup jL j
as n-1;
ð29Þ
0rsrt
sup jF^ j ðsÞF0 ðsÞj-P 0
as n-1,
ð30Þ
0rsrt
^ j , is defined by (7), resp. (8). j = 1,2,3, where F^ j ¼ 1S^ j , and S^ j , resp. L (b) If x 2 ð0, tF is such that (28) holds for all t ox, then (30) holds with t replaced by x. Using the martingale properties derived in Section 3 and Theorem 7, the proof of Theorem 8 is similar to that of Theorem 3.4.2 of Fleming and Harrington (1991). Note that condition (28) will hold under the conditions of Theorem 1.
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5. Simulation To explore the applicability of our results, a simulation was conducted based on 10,000 replications. Simulated data (T0i ) from a Weibull distribution with scale 10 and shape 2 were generated for sample sizes 100 and 200. These data were subjected to random right censoring by C0i having distribution G0 as Weibull with scale 10/9 and shape 2, giving about a 10% censorship rate. Then, the three progressive censoring schemes were applied to the Ti ¼ minðTi0 ,C0i Þ according to the construction in Section 3 at time points t = 2.5, 5 and 9. We specified the number of items to be removed under Scheme 2, namely R ¼ ðR1 ,R2 ,R3 Þ. Then, using Eqs. (9) and (10), we record the corresponding values of p ¼ ðp1 ,p2 ,p3 Þ and R ¼ ðR1 ,R2 ,R3 Þ for Schemes 1 and 3, respectively. A comparison to random right censoring (Scheme 4 with the same p) is also given. In each case, uniform confidence bands were calculated using (13). The percent reduction in the total time on test (TTT) is also given for each scheme. The results are summarized in Tables 1 and 2 for n = 100 and 200, respectively. In Table 1, we chose R to be (40,5,5), (5,40,5), (5,30,15) and (17,17,16) in order to study the effects of censoring either early, later or more evenly. Note that (5,30,15) was chosen over (5,5,40) because the latter resulted in too few items on trial after t= 9 for the asymptotic theory to hold. As can be seen, the coverage probabilities were all close to the nominal value of 95%. The average TTT if no progressive censoring took place was 781.2. The percent total time on test reduction was the greatest when R= (40,5,5) (early removal) and the least when R= (5,15,30), where 30 items were removed at t =9. This is consistent with the ‘‘new is better than used’’ property of the Weibull. The average band widths (W) together with the standard deviation of the widths (in parentheses) are given. W at t = 2 was 0.264 for each scheme and each value of R, since progressive censoring started after this time. The W values are quite comparable for each scheme. For example, when Scheme 2 with R ¼ ð40,5,5Þ is used, the widths of the Hall–Wellner confidence bands at times t =2, 4, 6.5, 9.5 were 0.264, 0.281, 0.330, 0.405, respectively. The coverage probability was 95.9%. The average TTT was 503.1 (a 35.6% reduction from 781.2). If we compare these results with a sample size of n= 50 to start and no progressive censoring, the widths would be have been 0.328, 0.375, 0.375, 0.376 at times t= 2, 4, 6.5, 9.5, respectively, with a coverage probability of 95.0%. The average TTT would have been 418.4 (smaller than 503.1). However, much narrower widths, at t = 2, 4, 6.5, were achieved by the progressive censoring of 50 items from 100, than one would get with an original sample size of 50. The larger width at t =9.5 was the result of very few items on test after t= 9. In Table 2, 100 items were progressively censored from n ¼ 200 in the same proportions as Table 1. As can be seen, the overall results are similar to those of Table 1. The average total time on test if no progressive censoring took place was 1746.5. For example, Scheme 2 with R= (80,10,10) (a total of 100 items removed), saw an average TTT of 1124.4 (35.6% reduction). The widths of the Hall–Wellner confidence bands at times t = 2, 4, 6.5, 9.5 were 0.191, 0.202, 0.240, 0.302, respectively. The coverage probability was 96.1%. If we compare the results with a sample size of n= 100 to start and no
Table 1 Simulation from a Weibull distribution (scale 10; shape 2) and progressive censoring at t = 2.5, 5 and 9. Various choices for R are specified. Average coverage probabilities (C.P. in %), reduction of the total time on test (%), and widths, W, (with std. dev.) of uniform confidence bands given by (13) at t= 2, 4, 6.5, 9.5 under the three progressive schemes and Scheme 4: iid random censoring. n =100
Scheme 1
Scheme 2
Scheme 3
Scheme 4
C.P. (%) TTT red. (%) W at 2 4 6.5 9.5
p ¼ ð0:43,0:12,0:24Þ 95.9 34.8 0.264 (0.036) 0.281 (0.008) 0.328 (0.015) 0.400 (0.022)
R ¼ ð40,5,5Þ 95.9 35.6 0.264 (0.036) 0.281 (0.008) 0.330 (0.017) 0.405 (0.026)
R ¼ ð54,39,16Þ 96.0 34.9 0.264 (0.036) 0.281 (0.008) 0.328 (0.015) 0.402 (0.026)
p ¼ ð0:43,0:12,0:24Þ 95.8 35.6 0.264 (0.036) 0.281 (0.009) 0.331 (0.021) 0.408 (0.036)
C.P. (%) TTT red. (%) W at 4 6.5 9.5
p ¼ ð0:05,0:56,0:29Þ 95.9 29.3 0.270 (0.001) 0.312 (0.023) 0.424 (0.036)
R ¼ ð5,40,5Þ 96.1 30.1 0.270 (0.001) 0.314 (0.026) 0.429 (0.043)
R ¼ ð89,32,13Þ 95.7 29.3 0.270 (0.001) 0.312 (0.024) 0.427 (0.041)
p ¼ ð0:05,0:56,0:29Þ 95.6 30.0 0.270 (0.001) 0.314 (0.028) 0.431 (0.053)
C.P. (%) TTT red. (%) W at 4 6.5 9.5
p ¼ ð0:05,0:42,0:67Þ 94.5 27.7 0.270 (0.001) 0.297 (0.012) 0.363 (0.052)
R ¼ ð5,30,15Þ 95.2 28.5 0.270 (0.001) 0.298 (0.013) 0.362 (0.058)
R ¼ ð89,42,8Þ 94.7 27.8 0.270 (0.001) 0.297 (0.013) 0.366 (0.054)
p ¼ ð0:05,0:42,0:67Þ 93.9 28.4 0.270 (0.001) 0.298 (0.015) 0.365 (0.061)
C.P. (%) TTT red. (%) W at 4 6.5 9.5
p ¼ ð0:18,0:26,0:69Þ 94.4 29.3 0.273 (0.002) 0.299 (0.009) 0.351 (0.048)
R ¼ ð17,16,17Þ 94.9 30.1 0.273 (0.002) 0.300 (0.010) 0.350 (0.053)
R ¼ ð77,46,8Þ 94.6 29.6 0.273 (0.002) 0.299 (0.010) 0.354 (0.054)
p ¼ ð0:18,0:26,0:69Þ 94.0 30.0 0.273 (0.002) 0.300 (0.012) 0.352 (0.057)
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Table 2 Simulation from a Weibull distribution (scale 10; shape 2) and progressive censoring at t = 2.5, 5 and 9. Various choices for R are specified. Average coverage probabilities (C.P. in %), reduction of the total time on test (%), and widths, W (with std. dev.), of the uniform confidence bands given by (13) at t = 2, 4, 6.5, 9.5 under the three progressive schemes and Scheme 4: iid random censoring. n= 200
Scheme 1
Scheme 2
Scheme 3
Scheme 4
C.P. (%) TTT red. (%) W at 2. 4. 6.5 9.5
p ¼ ð0:43,0:12,0:24Þ 96.2 35.2 0.191 (0.003) 0.202 (0.004) 0.239 (0.008) 0.300 (0.014)
R ¼ ð80,10,10Þ 96.1 35.6 0.191 (0.003) 0.202 (0.004) 0.240 (0.009) 0.302 (0.016)
R ¼ ð107,77,32Þ 96.0 35.2 0.191 (0.003) 0.202 (0.004) 0.239 (0.008) 0.300 (0.015)
p ¼ ð0:43,0:12,0:24Þ 96.2 35.6 0.191 (0.003) 0.202 (0.004) 0.240 (0.011) 0.303 (0.021)
C.P. (%) TTT red. (%) W at 4. 6.5 9.5
p ¼ ð0:05,0:56,0:29Þ 96.6 29.7 0.192 (0.000) 0.230 (0.012) 0.322 (0.023)
R ¼ ð10,80,10Þ 96.6 30.1 0.192 (0.000) 0.231 (0.013) 0.326 (0.028)
R ¼ ð177,64,25Þ 96.3 29.6 0.192 (0.000) 0.230 (0.013) 0.323 (0.024)
p ¼ ð0:05,0:56,0:29Þ 96.6 30.1 0.192 (0.000) 0.231 (0.014) 0.327 (0.032)
C. P. (%) TTT red. (%) W at 4. 6.5 9.5
p ¼ ð0:05,42,0:67Þ 95.7 28.1 0.192 (0.000) 0.215 (0.006) 0.283 (0.042)
R ¼ ð10,60,30Þ 94.8 28.5 0.192 (0.000) 0.216 (0.007) 0.289 (0.051)
R ¼ ð177,84,15Þ 95.5 28.2 0.192 (0.000) 0.215 (0.007) 0.283 (0.044)
p ¼ ð0:05,42,0:67Þ 95.4 28.5 0.192 (0.000) 0.216 (0.008) 0.285 (0.049)
C.P. (%) TTT red. (%) W at 4 6.5 9.5
p ¼ ð0:18,0:27,0:68Þ 95.5 30.0 0.194 (0.001) 0.216 (0.005) 0.274 (0.038)
R ¼ ð33,34,33Þ 94.6 30.1 0.195 (0.001) 0.217 (0.005) 0.273 (0.046)
R ¼ ð154,91,16Þ 95.2 29.8 0.194 (0.001) 0.216 (0.005) 0.274 (0.041)
p ¼ ð0:18,0:27,0:68Þ 95.2 30.1 0.195 (0.001) 0.217 (0.006) 0.275 (0.046)
progressive censoring (not shown in the tables), the widths would be have been 0.264, 0.269, 0.270, 0.273 at times t =2, 4, 6.5, 9.5, respectively. with a coverage probability of 95.7%. The average TTT was 781.2. Thus, much narrower widths were achieved by starting with 200 items on trial and the progressive censoring of 100 items of them, than one would get with an original sample size of 100, especially at smaller values of t. Thus, using progressive censoring for 200 items is a way to obtain narrower confidence bands at smaller t (as compared with starting with 100 items). The TTT is controlled by the systematic removal of items (1124.4 versus 1746.5), but is larger than the TTT of starting with 100 items and no progressive censoring (781.2).
Acknowledgements The author is grateful to the Executive and Associate Editors and to the referees for their insightful and helpful comments, which led to a substantial improvement of this paper. Appendix A. Proofs A.1. Proof of Theorem 7 Rt ^ ðuÞ, where As in the iid censoring case, the product limit estimator (7) can be derived recursively: S^ j ðtÞ ¼ 1 0 S^ j ðuÞ dL j ^ L j is defined by (8), and a bound on the bias can be found for Schemes 1–3. If S0 ðtÞ 40, then (Fleming and Harrington, 1991, Theorem 3.2.3 and Corollary 3.2.1) ( ) Z t^ S^ j ðtÞ S j ðuÞ dN 0j ðuÞ ¼ 1 dL0 ðuÞ , S0 ðtÞ Y 0j ðuÞ 0 S0 ðuÞ S^ j ðtÞS0 ðtÞ ¼ S0 ðtÞ
Z
t 0
Hj ðuÞ dM j ðuÞ þBj ðtÞ,
ð31Þ
where Hj ðuÞ ¼ S^ j ðuÞI½Y 0j ðuÞ 4 0=ðS0 ðuÞ Y 0j ðuÞÞ and Bj ðtÞ ¼ I½tjn o t
S^ j ðtjn ÞfS0 ðtjn ÞS0 ðtÞg , S0 ðtjn Þ
ð32Þ
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
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and tjn ¼ inffu : Y 0j ðuÞ ¼ 0g, j ¼ 1,2,3. The integral on the right side of (31) is a martingale, since the integrand is F ðjÞ n ðuÞ-predictable. Hence, it has expectation zero and (23) follows. To prove (24), we let Ej ¼ Ej ðtÞ ¼ ½Y 0j ðtÞ ¼ 0, j ¼ 1,2,3. If t r t1 , then none of the items are subject to censoring by P Scheme 1 before time t. Clearly, PðEj Þ ¼ f1SðtÞgn , j ¼ 1,2,3. Let N j ðuÞ ¼ ni¼ 1 I½Ti ru,Ti rCji . For t o tF , tk o t rtk þ 1 , 1 rk rm and mjk given by (16), P½Ej ¼
mj1 n X X ‘1 ¼ 0 ‘2 ¼ 0
mj,k1
X
P½N j ðt1 Þ ¼ ‘1 P½N j ðt2 Þ ¼ ‘ 2 jN j ðt1 Þ ¼ ‘1 P½N j ðtk Þ
‘k ¼ 0
¼ ‘ k jN j ðt1 Þ ¼ ‘1 , . . . ,N j ðtk1 Þ ¼ ‘ k1 P½Ej jN j ðt1 Þ ¼ ‘1 , . . . ,N j ðtk Þ ¼ ‘ k , P where ‘ k ¼ kw ¼ 1 ‘w and, with hw defined by (22), ! mj,k1 ðhk1 ðtk ÞÞ‘k ðSðtk Þ=Sðtk1 ÞÞmj,k1 ‘k ; P½N j ðtk Þ ¼ ‘ k jN j ðt1 Þ ¼ ‘1 , . . . , N j ðtk1 Þ ¼ ‘ k1 ¼ ‘k P½Ej jN j ðt1 Þ ¼ ‘1 , . . . ,N j ðtk Þ ¼ ‘ k ¼ ðhk ðtÞÞmjk :
ð33Þ
ð34Þ
Scheme j= 1: We use (33) and (34) with m1w ¼ dn1w ð1pw Þe and n1w ¼ dn1,w1 ð1pw1 Þe‘w and obtain P½E1 jN 1 ðt1 Þ ¼ ‘1 , . . . ,N 1 ðtk1 Þ ¼ ‘ k1 ¼
dn1,k1X ð1pk1 Þe
P½N 1 ðtk Þ ¼ ‘ k jN 1 ðt1 Þ ¼ ‘1 , . . . ,N 1 ðtk1 Þ ¼ ‘ k1 P½E1 jN 1 ðt1 Þ ¼ ‘1 , . . . ,N 1 ðtk Þ ¼ ‘ k
‘k ¼ 0
¼ ðq1k ðhk ðtÞÞÞ
dn1,k1 ð1pk1 Þe
r ðq1k ðhk ðtÞÞÞ
n1,k1 ð1pk1 Þ
:
Similarly, P½E1 jN 1 ðt1 Þ ¼ ‘1 , . . . ,N 1 ðtk2 Þ ¼ ‘ k2 r
dn1,k2X ð1pk2 Þe
P½N 1 ðtk1 Þ ¼ ‘ k1 jN 1 ðt1 Þ ¼ ‘1 , . . . ,N 1 ðtk2 Þ ¼ ‘ k2 ðq1k ðhk ðtÞÞÞn1,k1 ð1pk1 Þ
‘k1 ¼ 0
r ðq1,k1 3q1k 3hk ðtÞÞÞn1,k2 ð1pk2 Þ :
Continuing in this way, Eq. (24) follows and 0 og1 ðtÞ o 1. Scheme j= 2: The following lemma will be needed: Lemma 9. For any 0 o d o Sðtw Þ, let 1Fðtw Þ Sðtw Þ , hw ðxÞ ¼ xð1 þ dÞ ðxFðtw ÞÞ w ¼ 1,2, . . . ,m, where Fðtw Þ ox o 1. Then, there exists 0 o ew ¼ ew ðdÞ o1, such that, on ½ew ,1, hw is strictly increasing and hw ðxÞ o 1. P We use (33) and (34) with m2w ¼ nR w ‘ w and R k ¼ kw ¼ 1 Rw and obtain þ mX 2,k1 m2,k1 Fðtk ÞFðtk1 Þ ‘k Sðtk Þ ðm2,k1 ‘k Þ FðtÞFðtk Þ ðm2,k1 Rk ‘k Þ P½E2 jN 2 ðt1 Þ ¼ ‘ 1 , . . . ,N 2 ðtk1 Þ ¼ ‘ k1 r Sðtk1 Þ Sðtk1 Þ Sðtk Þ ‘k ‘k ¼ 0 m2,k1 Rk FðtÞFðtk1 Þ Sðtk Þ r : Sðtk1 Þ FðtÞFðtk Þ Using (33), it follows that Rw k Y Sðtw Þ PðE2 Þ rFðtÞn : FðtÞFðtw Þ w¼1 By assumption (A2), r k o 1, implies that du ð1r k Þ=r k 40 and " Sw #rw !n k Y Sðtw Þ PðE2 Þ r FðtÞ1 þ du : FðtÞFðtw Þ w¼1 By Lemma 9, there is a nðtÞ ¼ maxft,F 1 ðe1 ðduÞÞ, . . . , F 1 ðek ðduÞÞg, such that PðE2 Þ r g2 ðtÞ o1, where g2 is defined by (26). Scheme j = 3: We use (33) and (34) with m3w ¼ Y 03 ðtw Þ4Rw . Recall R‘ ¼ dnr ‘ e. Define n1 ðk1 Þ ¼ ðnk1 Þ4R1 , and nj ðk1 ,k2 , . . . ,kj Þ ¼ ðnj1 ðk1 ,k2 , . . . ,kj1 Þkj Þ4Rj . Similar to the case j ¼ 1, for tk o t rtk þ 1 , but t o tF , ! mX 3,k1 m3,k1 P½E3 jN 3 ðt1 Þ ¼ ‘1 , . . . ,N 3 ðtk1 Þ ¼ ‘ k1 ¼ ðhk1 ðtk ÞÞ‘k ðSðtk Þ=Sðtk1 ÞÞm3,k1 ‘k ðhk ðtÞÞm3k : ‘k ‘k ¼ 0
920
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
Note that rk ðm3,k1 ‘k Þ r m3,k and 0 o hk ðtÞ o 1, imply ðhk ðtÞÞm3k rðhk ðtÞÞrk ðm3,k1 ‘k Þ Hence, P½E3 jN 3 ðt1 Þ ¼ ‘1 , . . . ,N 3 ðtk1 Þ ¼ ‘ k1 ¼ ðq3k ðhk ðtÞÞÞm3,k1 : Continuing in this way, we obtain PðE3 Þ rðq31 3 . . . 3q3k ðhk ðtÞÞn . Since 0 o x o 1 implies q3w ðxÞ o 1, 1 r wr k, we have 0 og3 ðtÞ o 1. A.2. Proof of asymptotic equivalence (Theorem 1). We will assume that the censoring schemes are defined on the same probability space as in Section 3. We will use the following martingale inequality (Chung and Williams, 1990(1.3)): if M is a square integrable martingale and e 4 0,
ð35Þ P sup jMðsÞj 4 e r e2 EðMðtÞÞ2 : 0rsrt
In the proof of Theorem 1, we will use: Lemma 10. Let U1 ,U2 , . . . be independent uniform (0,1) random variables and let gn r n be a random sequence of positive integers, independent of the Ui sequence, such that gn -1, as n-1. If Fgn ðuÞ is the empirical distribution function of U1 ,U2 , . . . ,Ugn , then sup gn jFgn ðuÞuj ¼ OððnloglognÞ1=2 Þ,
a:s:
0rur1
¨ o+ and Re´ve´sz, 1981, Theorem 7.3.2), Gðu,nÞ, such that Proof. One can define a Kiefer process (Csorg pffiffiffiffiffi 1=2 sup j gn ðFgn ðuÞuÞgn Gðu, gn Þj-0 a:s:
ð36Þ
0rur1
¨ o+ and Re´ve´sz (1981) with aT ¼ T, we obtain as n-1. Applying Corollary 1.12.4 of Csorg lim
sup
jWðu,yÞj sup pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a:s: 1, 2TloglogT
ð37Þ
T-1 0 r y r T 0 r u r 1
where Wðu,yÞ,0 ru r 1,yZ 0, is a two-dimensional Brownian sheet. Since G has representation Gðu,yÞ ¼ Wðu,yÞuWð1,yÞ, we can replace W with G in (37). Combining this and (36), the lemma is proven. & Proof of Theorem 1. Consider the four censoring schemes constructed in Section 3. Recall that T1 ,T2 , . . . ,Tn are P independent random variables with common distribution function F. We have D ju ¼ ni¼ 1 Djui , ju ¼ 1,2,3, where Djui is defined by (19). Since each Djui of (19), is a martingale with respect to the s-field F n ðtÞ ( Corollary 6), so is D ju . By (35) " # sup jD ju ðtÞj Z e r e2 Eð½D ju ðtF Þ2 Þ,
P
ju ¼ 1,2,3:
ð38Þ
0 r t r tF
pffiffiffi To prove (11), it suffices to show that Eð½D ju ðtF Þ2 Þ r C n, for some constant C 40. From Corollary 6 n Z tF X 0 ðY0jui Y04i Þ2 dAi , Eð½D ju ðtF Þ2 Þ ¼ E/D ju ðtF Þ,D ju ðtF ÞS ¼ E i¼1
0
where A0i is defined by (14). Note that ðY0jui ðuÞY04i ðuÞÞ2 ¼ 1, if item i was censored before u by one Scheme (ju or 4) and is observed to be functioning at time u under the other Scheme (4 or ju), and = 0, otherwise. For o 2 O, the underlying probability space, let tjuiu ðoÞ ¼ inffu : jY0jui ðu, oÞY04i ðu, oÞj ¼ 1g. If Y0jui ðu, oÞ ¼ Y04i ðu, oÞ, for all u, put tjuiu ðoÞ ¼ tF . Hence, tjuiu is a discrete random variable with possible values tF and tk, 1 rk r m. Thus, Z tF Z tF 0 0 ðY0jui Y04i Þ2 dAi ¼ jY0jui Y04i j dAi rA0i ðtF ÞA0i ðtjui u Þ: tj00 i
0
We have EfA0i ðtF ÞA0i ðtjuiu Þjtjuiu g ¼ I½tjuiu o tF , a:s:, since ½tjuiu ¼ tk D ½tk oTi0 and EfA0i ðtF ÞA0i ðtk Þ jtk o Ti0 g ¼ 1. Consequently, we obtain u Þ ð39Þ Eð½D ju ðtF Þ2 Þ r Eðujun Pn u ¼ i ¼ 1 I½tjui u o tF is the number of items censored by one scheme and not the other at some time in the where ujun experiment. Put u ¼ ujun
m X k¼1
ujuk ,
ð40Þ
M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
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P ju ¼ 1,2,3, where ujuk ¼ ni¼ 1 I½tjuiu ¼ tk is the number of items observed functioning at time tk by both Schemes (ju and 4) and censored at tk by one scheme and not the other. Note that sup jY 0ju ðtÞY 04 ðtÞj rvujun ,
t r tF
ju ¼ 1,2,3
ð41Þ
Let FnðkÞ denote the empirical distribution function of fUiðkÞ : i 2 rju ðtk , oÞ \ r4 ðtk , oÞg, FnðkÞ denote the empirical distribution j0 4k j0 k function of fUiðkÞ : i 2 rju ðtk , oÞg, and let nju4k ¼ ðrju ðtk , oÞ \ r4 ðtk , oÞÞ; ju ¼ 1,2,3, 1 r kr m. Recall the notation (16). Scheme ju ¼ 1: We will prove (11) and (12) together. Recall the notation from Section 3. Note that none of the items were ðp1 Þbn11 p1 c=n11 j r n11 jFnð1Þ ðp1 Þp1 jþ 1. removed before time t1. Hence r1 ðt1 , oÞ \ r4 ðt1 , oÞ ¼ r1 ðt1 , oÞ and u11 n11 jFnð1Þ 11 11 Using Schwarz’s inequality and noting that the variance of the binomial, given n11 and p1 is n11p1(1 p1), we obtain pffiffiffi n 2 1=2 : ð42Þ Eu11 r 1 þfEn211 jFnð1Þ ðp Þp j g r 1þ 1 1 11 2 By Lemma 10, as n-1, u11 ¼ OððnloglognÞ1=2 Þg, a:s. Consider time tk. Under Schemes 1 and 4 ðkÞ ðkÞ ðkÞ ðkÞ u1k n14k jFnðkÞ ðpk ÞFnðkÞ ðUðbn Þj r n14k jFnðkÞ ðpk Þpk j þ n14k jFnðkÞ ðUðbn ÞUðbn jþ n14k jUðbn pk j 14k 14k 14k 14k 1k pk cÞ 1k pk cÞ 1k pk cÞ 1k pk cÞ
ð43Þ
ðkÞ is the bn1k pk c th-order statistic from fUiðkÞ : i 2 r1 ðtk , oÞg. Similar to (42), the expectation of the first term of where Uðbn 1k pk cÞ pffiffiffi ðkÞ ðkÞ , n jF ðkÞ ðU ðkÞ ÞUðbn j is binomial and hence (43) is bounded by n=2. For the second term, given n14k and Uðbn 1k pk cÞ pffiffiffi 1k pk cÞ 14k n14k ðbn1k pk cÞ ðkÞ , the expected value of the second term of (43) is bounded by n=2. For the third term of (43), the variance of n14k Uðbn 1k pk cÞ given n14k and n1k , is
n214k pk ð1pk Þn21k ðn1k þ 1Þ2 ðn1k þ2Þ
r npk ð1pk Þ:
Hence, again by Schwarz’s inequality, pffiffiffi ðkÞ n bn1k pk c n14k ðkÞ U þ 1, r p j r En En14k jUðbn þE k 14k ðbn p cÞ p cÞ 1k k 1k k 2 n1k þ1 n1k þ1
ð44Þ
pffiffiffi pffiffiffi Thus, Eu1k r 3 n=2þ 1, and Euu1n r ð3m2Þ=2 n þ m. Consequently, (11), for ju ¼ 1, follows from (38), (39) and (40). By Lemma 10, the first two terms on the right side of (43) equal OðnloglognÞ1=2 , a:s. For the third term, ðkÞ ðkÞ ðkÞ n14k jUðbn pk j r n14k jUðbn FnðkÞ ðUðbn Þj þ1 ¼ OððnloglognÞ1=2 Þ 1k 1k pk cÞ 1k pk ccÞ 1k pk cÞ
a:s:
ð45Þ
again by Lemma 10. Hence, for each k r m, u1k ¼ OððnloglognÞ1=2 Þ, a:s: as n-1. Consequently, (12) for ju ¼ 1, follows from this and (41). Scheme ju ¼ 2: We first prove (11). Note that, under the conditions of our Theorem, P½Y 02 ðtÞ ¼ 0,i:o: ¼ 0, tm ot o tF , by Theorem 7. This implies m2k -1, a:s:, 1 rk rm. For time t1, R1 4n21 ðp1 Þ ðp1 Þp1 j þjp1 n21 np1 Sðt1 Þj þ n21 ð1p1 ÞI½n21 o R1 þ 1: r n21 jFnð1Þ u21 n21 Fnð1Þ 21 21 n21 pffiffiffi As in the ju ¼ 1 case, the expectation of the first term of the right side of the above inequality is bounded by n þ 1. The pffiffiffi second term is p1 times the centred binomial with parameters n and S(t1) and hence its expectation is bounded by n þ 1. For the third term, P½n21 o R1 rP½nnp1 Sðt1 Þ r nn21 r P½nð1p1 ÞSðt1 Þ rðnn21 nFðt1 ÞÞ rexpf2nð1p1 Þ2 Sðt1 Þ2 g,
ð46Þ
by Hoeffding’s inequality (Petrov, 1995, p. 78). Consequently, P½n21 oR1 , i:o: ¼ 0 and the expectation of the third term is O(1). At time tk, we have ðkÞ ðkÞ ðkÞ ðkÞ u2k n24k jFnðkÞ ðpk ÞFnðkÞ ðUðR Þj rn24k jFnðkÞ ðpk Þpk jþ n24k jFnðkÞ ðUðR ÞUðR jþ n24k jUðR pk j, 24k 24k 24k 24k k 4n2k Þ k 4n2k Þ k 4n2k Þ k 4n2k Þ
ð47Þ
ðkÞ is the ðRk 4n2k Þ th order statistic from the set r2 ðtk , oÞ of n2k elements. As in the case of ju ¼ 1 above, the first where UðR k 4n2k Þ pffiffiffi two terms of the right side of (47) have expectations bounded by n þ 1. The third term is bounded by ðkÞ pk jI½n2k Z Rk : n24k ð1pk ÞI½n2k o Rk þ n24k jUðR kÞ
ð48Þ
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M.D. Burke / Journal of Statistical Planning and Inference 141 (2011) 910–923
Since ½n2k oRk D½Y 02 ðtÞ ¼ 0, where tk ot r tk þ 1 , and P½Y 02 ðtÞ ¼ 0 r fg2 ðtÞgn , 0 o g2 ðtÞ o 1, (Theorem 7), we obtain for the first term of (48): P½n2k o Rk , i:o: ¼ 0 and En24k ð1pk ÞI½n2k o Rk ¼ Oð1Þ. The second term of (48) is bounded by ðkÞ R R ðkÞ n24k jUðR pk j r n24k UðR k þ n24k k pk : ð49Þ kÞ kÞ n2k n2k pffiffiffi As with (44) and (45), the first term of (49) has expected value bounded by n=2þ 1. For the second term of (49), (see (9)) ! ! k1 k1 X X R bnrk Sðtk Þc n n n pk ¼ 24k pk n 1 r‘ Sðtk Þ 2k r n 1 r‘ Sðtk Þ 2k n24k k pk ¼ n24k n2k n2k n2k n n ‘¼1 ‘¼1 ! k 1 X n Sðtk Þ m2,k1 Sðtk Þ m2,k1 þ n 1 r‘ Sðtk Þ, r n 2k Sðtk1 Þ n m2,k1 Sðtk1 Þ n ‘¼1
where m2,k1 ¼ n2,k1 Rk1 ¼ n2,k1 bnpk1 Sðtk1 Þc. So, given n2,k 1, n2k is a binomial random variable with parameters m2,k 1 and S(tk)/S(tk 1). By Schwarz’s inequality, the first term on the right side of the above inequality has expected value pffiffiffi bounded by n=2. The second term on the right side is bounded by
n
! k2 X Sðtk Þ n2,k1 1 r‘ Sðtk1 Þ þ1: Sðtk1 Þ n ‘¼1
pffiffiffi pffiffiffi By induction, the second term has expectation bounded by n=2. Consequently, Eu2k r 6 n þ5. Hence by (38)–(40), (11) is proven (ju ¼ 2). We will prove (12) for ju ¼ 2, by showing supt jn1 Y 02 ðtÞpðtÞj ¼ Oðn1=2 ðloglognÞ1=2 Þ, a.s. directly. For 0 rt r t1 , P (no censoring has taken place), Y 02 ðtÞ ¼ ni¼ 1 I½Ti Z t. Hence by the law of the iterated logarithm for the empirical process, sup jn1 Y 02 ðtÞpðtÞj ¼ Oðn1=2 ðloglognÞ1=2 Þ
a:s:
ð50Þ
0 r t r t1
P þ For t1 o t rt2 , we can write Y 02 ðtÞ ¼ n21 m21 Em21 ðtÞ, where Em21 ðtÞ ¼ ðm21 Þ1 i2r2 ðt1 þ Þ I½t1 o Ti r t is the empirical distribution function of the m21 lifetimes Ti that are functioning and not censored at t1. These Ti have (conditional) survival function S(t)/S(t1). Then, jn1 Y 02 ðtÞpðtÞj ¼ n1 jY 02 ðtÞnð1r1 ÞSðtÞj rn1 jm21 nð1r1 ÞSðt1 Þj þn1 m21 jðEm21 ðtÞðSðt1 ÞSðtÞÞ=Sðt1 ÞÞj þ n1 jðnð1r1 ÞSðt1 Þm21 ÞðSðt1 ÞSðtÞÞ=Sðt1 Þj: Thus, sup0 r t r t2 jn1 Y 02 ðtÞpðtÞj ¼ Oðn1=2 ðloglognÞ1=2 Þ, a:s: by (50) and Lemma 10, respectively. By induction, ðloglognÞ1=2 Þ, a:s:. It is well known that sup0 r t r tF jn1 Y 04 ðtÞpðtÞj ¼ Oðn1=2
sup0 r t r tF jn1 Y 02 ðtÞpðtÞj ¼ Oðn1=2
ðloglognÞ1=2 Þ, a:s: Hence (12) is proven for ju ¼ 2. Scheme ju ¼ 3: As in the Scheme 2 case R1 4n31 ðp Þ ðp1 Þp1 j þ ð1p1 Þjn31 nSðt1 Þj þ R1 I½n31 oR1 þ 1, rn31 jFnð1Þ u31 n31 1Fnð1Þ 1 31 31 n31 pffiffiffi and a (46)-type inequality holds for P½n31 oR1 . Consequently, Eðu31 Þ ¼ Oð nÞ and u31 ¼ OððnloglognÞ1=2 Þ, a:s. At time tk, we have ðkÞ ðkÞ ðkÞ ðkÞ ðpk ÞFnðkÞ ðUðn Þj rn34k jFnðkÞ ðpk Þpk jþ n34k jFnðkÞ ðUðn ÞUðn j þ n34k jUðn pk j, u3k n34k jFnðkÞ 34k 34k 34k 34k 3k R 4n3k Þ 3k R 4n3k Þ 3k R 4n3k Þ 3k R 4n3k Þ k
k
k
k
ð51Þ ðkÞ is the ðn3k Rk 4n3k Þth order statistic from the set r3 ðtk , oÞ of n3k elements. As above, the first two terms where Uðn 3k Rk 4n3k Þ pffiffiffi of (51) are OððnloglognÞ1=2 Þ, a:s: and have expectations bounded by n þ 1. The third term of (51) is bounded by ðkÞ pk j. Similar to the ju ¼ 2 case, condition (A3) implies that rk ¼ Eðn3k Þð1pk Þ, 0 o pk o1 and Rk I½n3k o Rk þ n34k jUðn 3k Rk 4n3k Þ pffiffiffi P½n3k o Rk , i:o: ¼ 0 and EðRk I½n3k o Rk Þ ¼ Oð1Þ. Similarly, we also obtain Eu3k ¼ Oð nÞ and u3k ¼ OððnloglognÞ1=2 Þ, a:s: (11)
and (12) follows for Scheme 3.
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923
A.3. Proof of Theorem 2 We first prove pffiffiffi n sup jF^ ju ðsÞF^ 4 ðsÞj-P 0,
ð52Þ
0rsrt
ju ¼ 1,2,3. We have Eq. (31), for 0 rs r t, j ¼ 1,2,3,4, where M j is defined by (6), Hj ðuÞ is F nðjÞ ðuÞ-predictable and pffiffiffi nsup0 r s r t jBj ðsÞj ¼ oP ð1Þ, as n-1, by (32) and Theorem 7 (Fleming and Harrington, 1991, Proof of Theorem 6.3.1). Hence for ju ¼ 1,2,3, Z s Z s pffiffiffi pffiffiffi pffiffiffi njF^ ju ðsÞF^ 4 ðsÞj rS0 ðsÞ Hju ðuÞd nM ju ðuÞ H4 ðuÞd nM 4 ðuÞ þ oP ð1Þ: 0
0
By (35) and Corollary 5, for j ¼ 1,2,3,4, Z s
Z t pffiffiffi Hj ðuÞðnpðuÞÞ1 d nM j ðuÞ 4 e r En1 e2 ðnHj ðuÞðpðuÞÞ1 Þ2 dA j ðuÞ: P sup 0rsrt
0
ð53Þ
0
Using (12) and the fact that supu r t jn1 Y 04 ðuÞpðuÞj-P 0, we obtain sup0 r u r t jn1 Y 0ju ðuÞpðuÞj-P 0, ju ¼ 1,2,3 and Z t ðnHj ðuÞðpðuÞÞ1 Þ2 dA j ðuÞ r n1 A j ðtÞ sup ðnHj ðuÞðpðuÞÞ1 Þ2 ¼ oð1Þ a:s: n1 e2 0
0rurt
Since the above is uniformly bounded (in n) (Billingsley, 1995, Theorem 16.5), (53) converges to 0 as n-1. Finally, using the integration by parts formula (Fleming and Harrington, 1991, Theorem A.1.2), Z s pffiffiffi 1 1 2 sup d nðM ju ðuÞM 4 ðuÞÞ r pffiffiffi sup jD ju ðsÞj 1 : pðtÞ n0 r s r t 0 r s r t 0 npðuÞ By Theorem 1, the above supremum converges to 0, in probability and hence (52) is proven. Consider 2 nI½Y ðuÞ 40 sup0 r u r t jS^ j0 ðuÞS20 ðuÞj 1 1 0ju sup jðHju ðuÞÞ2 Y 0ju ðuÞðpðuÞÞ1 j r sup þ : S0 ðtÞ 0 r u r t pðuÞ S20 ðtÞpðtÞ Y 0ju ðuÞ 0rurt
ð54Þ
By (12), (29), (30) and the fact that n1 sup0 r u r t jY 04 ðuÞpðuÞj-P 0, we obtain that the left side of (54) converges to 0, in Rs probability. It follows that 0 ðHju ðuÞÞ2 Y 0ju ðuÞdL0 ðuÞ-P vðsÞ, and Theorem 2 is proven. & References Andersen, P.K., Borgan, Ø., Gill, R.D., Keiding, N., 1993. Statistical Models Based on Counting Processes. Springer-Verlag, New York. Bagdonavicˇius, V., Nikulin, M., 2002. Accelerated Life Models: Modeling and Statistical Analysis. Chapman & Hall/CRC, Boca Raton, Fl. ¨ Balakrishnan, N., Aggarwala, R., 2000. Progressive Censoring: Theory, Methods, and Applications. Birkhauser, Boston. Billingsley, P., 1995. Probability and Measure, third ed. Wiley, New York. Bordes, L., 2004. Non-parametric estimation under progressive censoring. J. Statist. Plann. Inference 119, 171–189. + S., Horva´th, L., 1981. Strong approximations of some biometric estimates under random censorship. Z. Wahrsch. verw. Geb. 56, ¨ o, Burke, M.D., Csorg 87–112. ¨ Chung, K.L., Williams, R.J., 1990. Introduction to Stochastic Integration, second ed. Birkhauser, Boston. + S., Horva´th, L., 1986. Confidence bands for censored samples. Canad. J. Statist. 14, 131–144. ¨ o, Csorg + M., Re´ve´sz, P., 1981. Strong Approximations in Probability and Statistics. Academic Press, New York. ¨ o, Csorg Fleming, T.R., Harrington, D.P., 1991. Counting Processes and Survival Analysis. Wiley, New York. Gill, R.D., 1980. Censoring and stochastic integrals, Mathematical Centre Tracts, vol. 124, Mathematisch Centrum, Amsterdam. Gillespie, M., Fisher, L., 1979. Confidence bands for the Kaplan–Meier survival curve estimates. Ann. Statist. 7, 920–924. Hall, W.J., Wellner, J.A., 1980. Confidence bands for a survival curve from censored data. Biometrika 67, 133–143. Petrov, V.V., 1995. Limit Theorems of Probability Theory. Oxford University Press, Oxford.