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Some properties of classes of life distributions with unknown age
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Statistics & Probability Letters 69 (2004) 333–342 www.elsevier.com/locate/stapro
Some properties of classes of life distributions with unknown age Ibrahim A. Ahmad Department of Statistics and Actuarial Science, University of Central Florida, Orlando, FL 32816-2370, USA Received 4 March 2004; received in revised form 3 April 2004 Available online 8 July 2004
Abstract Some properties of the used better than aged (UBA) and the used better than aged in expectation (UBAE) classes of life distributions are given. These properties include moment inequalities and moment generating functions behavior. In addition, nonparametric estimation and testing of the survival functions of these classes are discussed. r 2004 Elsevier B.V. All rights reserved. Keywords: Used better than aged life distributions; Moment inequalities; Moment generating function; Estimation and life testing; Consistency and asymptotic normality; Pitman’s efficacy
1. Introduction Let X X0 be a random life with distribution function ðdfÞF and a survival function ðsfÞ F ¼ 1 F. At age t, define the random residual life by X tR whose sf is given by 1 dx. The mean of F ðx þ tÞ=FðtÞ; x; tX0. Assume that X has a finite mean m ¼ EðX Þ ¼ 0 FðxÞ X t is called mean residual life (MRL) and is given by Z 1 du=F ðtÞ; tX0; FðtÞ40: mðtÞ ¼ EðX t Þ ¼ FðuÞ ð1:1Þ t
Corresponding author. Tel.: +1-407-823-2289.
E-mail address: [email protected] (I.A. Ahmad). 0167-7152/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2004.06.029
ARTICLE IN PRESS I.A. Ahmad / Statistics & Probability Letters 69 (2004) 333–342
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Further, the hazard rate of X is defined by hðtÞ ¼
d ¼ f ðtÞ ; ln FðtÞ dt FðtÞ
tX0; F ðtÞ40;
where f ðtÞ ¼ F 0 ðtÞ is the probability density of X assuming it exists. Note that if limt!1 hðtÞ ¼ hð1Þ exists and is positive, then cf. Willmot and Cai (2000) mð1Þ ¼ lim mðtÞ ¼ t!1
1 : hð1Þ
ð1:2Þ
Two classes of life distributions were introduced by Alzaid (1994) who called them the used better than aged (UBA) and the used better than aged in expectation (UBAE) classes. Precisely we have the following definitions: Definition 1.1. The df F is said to be used better than aged (UBA) if 0omð1Þo1 and for all x; tX0, þ tÞXF ðxÞet=mð1Þ : Fðx
ð1:3Þ
Definition 1.2. The df F is said to be used better than aged in expectation (UBAE) if 0omð1Þo1 and for all tX0, mðtÞXmð1Þ:
ð1:4Þ
Alzaid (1994) showed that the UBA class is a subclass of UBAE and that if F is IHR (increasing hazard rate), then F is UBA. Similar implications between UBAE, NBUE (new is better than used in expectation) and HNBUE (harmonic new is better than used in expectation) were given by Di Crescenzo (1999). More recently, Willmot and Cai (2000) showed that the UBA class includes the DMRL class (decreasing mean residual lifetime) while the UBAE includes the DVRL (decreasing variance residual lifetime). We thus have IHR DMRL UBA UBAE [ DVRL For definitions and details of the classes IHR, NBUE and DMRL, see Barlow and Proschan (1981) while for the HNBUE, see Klefsjo (1983). The DVRL class means that VarðX t Þ is decreasing in t, as t ! 1. The present work addresses probability and inferential properties of the UBA and UBAE classes. These properties seem to have not been dealt with by authors discussing these two classes. In Section 2, we give moment inequalities of these two classes. In Section 3, we give upper bounds of the moment generating functions in the two classes that guarantee their existence and finiteness. In Section 4, we estimate FðxÞ whenever assumed to be UBA (or UBAE) and demonstrate its consistency with rate. Finally, testing in these classes is discussed in Section 5.
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2. Moments inequalities The following result gives moments inequality for the UBA class. Theorem 2.1. Let F be UBA such that for some integers r; sX0; mðrþsþ2Þ ¼ EðX rþsþ2 Þo1. If lðrÞ ¼ mðrÞ =r!; rX0, then lðrþsþ2Þ Xlðrþ1Þ ½mð1Þ sþ1 :
ð2:1Þ
Proof. Since F is UBA, we have þ tÞXF ðxÞet=mð1Þ : Fðx
ð2:2Þ
Thus for all integers r; sX0, Z 1Z 1 Z xr ts F ðx þ tÞ dx duX 0
0
1
xr F ðxÞ dx
Z
0
ts et=mð1Þ dt:
ð2:3Þ
0
Now, the right-hand side (2.3) is equal to Z 1 Z sþ1 sþ1 r ½mð1Þ s! x F ðxÞ dx ¼ ½mð1Þ s!E 0
1
X
xr dx ¼
0
mðrþ1Þ s!½mð1Þ sþ1 : rþ1
ð2:4Þ
Also, the left-hand side of (2.3) is equal to Z 1Z u Z 1 Z 1 r s rþsþ1 ðu vÞ v FðuÞ dv du ¼ ws ð1 wÞr dw F ðuÞu 0 0 0 0 Z 1 urþsþ1 F ðuÞ du ¼ bðr þ 1; s þ 1Þ 0
mðrþsþ1Þ r!s! ¼ : ðr þ s þ 1Þ ðr þ s þ 1Þ!
(2.5)
The result follows from (2.4) and (2.5). Remark 2.1. Theorem 2.1 above may be extended as follows: The definition of the UBA is equivalent to the following: let x1 ; . . . ; xkþ1 be nonnegative, then F is UBA, if and only if ! kþ1 Pkþ1 X xi XF ðx1 Þe 2 xi =mð1Þ : ð2:6Þ F i¼1
Using the same methodology, we thus have: If F is UBA then Pkþ1 lPkþ1 r þkþ1 Xlðr1 þ1Þ ½mð1Þ 2 ri þk : i¼1
i
Corollary 2.1. Let r ¼ s ¼ 0, then mð2Þ X2mð1Þ mð1Þ. Corollary 2.2. Let s ¼ 0, then mðrþ2Þ Xðr þ 2Þmðrþ1Þ mð1Þ. Corollary 2.3. Let r ¼ 0, then mðsþ2Þ Xðs þ 2Þ!mð1Þ ½mð1Þ sþ1 .
ð2:7Þ
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Next, we give analogous result for the UBAE class. Toward this end we define the equilibrium life time X~ whose sf is defined by Z 1 1 du: FðuÞ ð2:8Þ F ð1Þ ðxÞ ¼ mð1Þ x Willmot and Cai (2000) showed that F is UBAE, if and only if, F ð1Þ is UBA. Using this result, we can prove the following theorem. Theorem 2.2. Let F be UBAE such that for some integers r; sX0, mðrþsþ3Þ ¼ EðX rþsþ3 Þo1. Then lðrþsþ3Þ Xlðrþ2Þ ½mð1Þ sþ1 :
ð2:9Þ
Proof. Let m~ ðrÞ denote the rth moment of X~ . Hence m~ ðrÞ ¼ mðrþ1Þ =ðr þ 1Þmð1Þ ; note that m~ ðrÞ
Z 1 Z 1 r r1 ¼r x F ð1Þ ðxÞ dx ¼ x F ðuÞ du dx mð1Þ 0 0 x Z u Z 1 Z 1 r 1 ¼ xr1 dx du ¼ xr F ðxÞ dx F ðuÞ mð1Þ 0 m 0 ð1Þ 0 Z X mðrþ1Þ 1 E xr dx ¼ : ¼ mð1Þ ðr þ 1Þmð1Þ 0 Z
ð2:10Þ
1
r1
But, F is UBAE, then F ð1Þ is UBA and thus using Theorem 2.1, m~ ðrþsþ2Þ m~ ðrþ1Þ X ½mð1Þ sþ1 ; ðr þ s þ 2Þ! ðr þ 1Þ!
(2.11)
ð2:12Þ
since m~ 1 ð1Þ is the same as mð1Þ. The result follows in light of (2.10). Remark 2.2. The above result admits the following generalization: Pkþ1 lðPkþ1 r þkþ2Þ Xlðr1 þ2Þ ½mð1Þ 2 ri þk : i¼1
ð2:13Þ
i
We can also state some interesting special cases: (i) r ¼ s ¼ 0 : m3 X3mð2Þ mð1Þ. (ii) s ¼ 0 : mðrþ3Þ Xðr þ 3Þmð1Þ. mð2Þ ½mð1Þ sþ1 . (iii) r ¼ 0 : mðsþ3Þ Xðs þ 3Þ! 2 Remark 2.3. Several authors derived moments inequalities of different families of life distributions such as IHR, IHRA, NBU, NBUE, DMRL and HNBUE as well as studied the behavior of lmþn =ln as n ! 1 for all mX1, cf. Ahmad (2001), Ahmad and Mugdadi (2004), Fagiuoli and Pellerey (1997) and the references therein. Thus the above results contribute to this literature for the UBA and UBAE classes.
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3. Existence of moment generating functions In this section, we show that the moment generating function (mgf) of X exists and is finite for the UBA class if mð1Þ exists while in the UBAE class we need mð2Þ to exist. Actually, upper bounds of the mgf’s are given. Precisely, we have Theorem 3.1. If F is UBA and if mð1Þ o1, then for all nonnegative yaðmð1ÞÞ1 , jðyÞp1 þ
mð1Þ y ; 1 ymð1Þ
ð3:1Þ
where jðyÞ ¼ EðeyX Þ. Proof. Note that
Z
jðyÞ ¼ 1 þ y
1
eyx F ðxÞ dx;
ð3:2Þ
0
for
Z
1
1þy
e F ðxÞ dx ¼ 1 þ yE yx
0
0
eyx dx ¼ 1 þ ðEðeyX 1ÞÞ ¼ jðyÞ: 0
Since F is UBA, we have Z Z 1Z 1 eyx F ðx þ tÞ dx dtX 0
Z
1
dx eyx FðxÞ 0
Z
1
et=mð1Þ dt:
ð3:3Þ
0
The left-hand side of (3.3) is equal to Z 1
Z 1Z u 1 yðuvÞ yu e e F ðuÞ du mð1Þ F ðuÞ dv du ¼ y 0 0 0 mð1Þ 1 ¼ 2 ðjðyÞ 1Þ : y y The right-hand side of (3.3) is equal to Z 1 dx ¼ mð1Þ ðjðyÞ 1Þ: eyx FðxÞ mð1Þ y 0
(3.4)
ð3:5Þ
The result now follows from (3.4) and (3.5). Theorem 3.2. If F is UBAE and if mð2Þ o1, then for all nonnegative yaðmð1ÞÞ1 , jðyÞp1 þ
mð1Þ y þ y2 ðmð2Þ mð1Þmð1Þ Þ : 1 mð1Þy
ð3:6Þ
Proof. Since F is UBAE then F ð1Þ is UBA and if j~ ð1Þ is the mgf of F ð1Þ then we have ~ ð1Þ ðyÞp1 þR m~ ð1Þ y=1 ymð1Þ. But m~ ð1Þ ¼ mð2Þ =2mð1Þ and j ~ ð1Þ ðyÞ ¼ 1=mð1Þ yðjðyÞ 1Þ, note that j 1 ~ ð1Þ ðyÞ ¼ y 0 eyx F ð1Þ ðxÞ dx þ 1. Thus the result follows from the relation j Z 1 1 1 1 eyx F ð1Þ ðxÞ dx ¼ jðyÞ : ð3:7Þ 2 2 y mð1Þ y mð1Þ y 0
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Hence, 1 mð1Þ y
ðjðyÞ 1Þp1 þ
mð2Þ y : 2mð1Þ ð1 ymð1ÞÞ
ð3:8Þ
The result now follows. It follows from the above results that if F is UBA and if mð1Þ o1 then all moments of F exist and are finite. When F is UBAE we need mð2Þ o1.
4. Estimating UBA survival functions In this section, we address the questions of estimating F ðxÞ given that it is UBA. The case of the UBAE class will be dealt with in a sequel work. Let X 1 ; . . . ; X n denote aP random sample from a life distribution F. With no restriction on F , the empirical df F n ðxÞ ¼ 1=n ni¼1 Iðx X i Þ; IðxÞ ¼ 1 if xX0 and is 0 otherwise, is a widely used nonparametric estimate of F ðxÞ. When F is UBA, we shall modify F n ðxÞ and offer an estimate that is UBA. To this end we need the following simple result.
þ tÞet=mð1Þ . Then F ¼ 1 F is UBA. Lemma 4.1. Let F be a df and define F ðxÞ ¼ inf t40 Fðx
Proof. Note that
þ y þ tÞet=mð1Þ ¼ inf F ðx þ y þ tÞeðyþtÞ=mð1Þ ey=mð1Þ F ðx þ yÞ ¼ inf Fðx t
t40
¼ inf F ðx þ uÞe
u=mð1Þ y=mð1Þ
u4y
e
X inf F ðx þ uÞeu=mð1Þ ey=mð1Þ ¼ F ðxÞ ey=mð1Þ : u40
We then propose to estimate FðxÞ by,
F n ðxÞ ¼ inf F n ðx þ tÞet=mð1Þ : t40
ð4:1Þ
Note that F n ðxÞ is UBA and also that for computational needs one can write
F n ðxÞ ¼ min F n ðx þ X ðiÞ ÞeX ðiÞ =mð1Þ ; 1pipn
ð4:2Þ
where X ð1Þ ; . . . ; X ðnÞ are the order statistics. Note also that the estimate in (4.1) or (4.2) assumes that mð1Þ is known. Let us address this case first, then will show how to deal with some of the
cases when mð1Þ is unknown. To show the consistency of F n ðxÞ and its rate, the following lemma is essential. Lemma 4.2. Let Dn ¼ supx jF n ðxÞ F ðxÞj. Then for any xX0,
jF n ðxÞ F ðxÞjp3Dn :
ð4:3Þ
Proof. Note that
jF n ðxÞ F ðxÞjpjF n ðxÞ F n ðxÞj þ Dn :
ð4:4Þ
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But, F n ðxÞpF n ðxÞ for all x we see that
jF n ðxÞ F n ðxÞj ¼ F n ðxÞ F n ðxÞ ¼ inf ½F n ðxÞ F n ðx þ tÞet=mð1Þ t40 t=mð1Þ
¼ inf f½F n ðxÞe
F n ðx þ tÞ et=mð1Þ g
t40
¼ inf fF n ðxÞet=mð1Þ F ðxÞet=mð1Þ þ F ðxÞet=mð1Þ F n ðx þ tÞ t40
þ F ðx þ tÞ F ðx þ tÞg ðF n ðx þ tÞ F ðx þ tÞÞ ¼ inf fet=mð1Þ ðF n ðxÞ FðxÞÞ t40
t=mð1Þ þ tÞ FðxÞe Þg ðFðx
p2Dn :
(4.5)
The result follows from (4.4) and (4.5). Theorem 4.1. (i) Let rX1 be an integer, then r=2 9
r Gðr=2Þnr=2 EjF n ðxÞ FðxÞj pr 2
(ii) jF ðxÞ F ðxÞj ¼ OWPI ðn1=2 ðln ln nÞ1=2 Þ. n
r EðDrn Þ. But we easily see that using Proof. (i) Note that from Lemma 4.2, EjF n ðxÞ FðxÞjp3 Corollary 3 of Massart (1990), we have Z 1 Z 1 r r1 EDn ¼ r x PðDn 4xÞ dx ¼ r xr1 Pðn1=2 Dn 4n1=2 xÞ dx 0 0 Z 1 pffiffiffi r=2 ur1 Pð nDn uÞ du ¼ rn 0
p2rnr=2
Z 0
1
2
ur1 e2u du ¼
rnr=2 2r=2
Gðr=2Þ:
(4.6)
The result is now immediate. (ii) Follows directly from Lemma 4.2 and the standard law of iterated logarithm of Dn , cf. Karr (1993, p. 207). In the following result, we give a rate of convergence in the moment generating function of
jF n ðxÞ F ðxÞj. Theorem 4.4. For any y40, pffiffi pffiffiffiffiffiffi 2 Eey njF n ðxÞF ðxÞj p1 þ 3 2pye18y :
Proof. Using Lemma 4.2, we get that pffiffi pffiffi Eey njF n ðxÞF ðxÞj pEe3y nDn :
ð4:7Þ
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Note that using Corollary 3 of Massart (1990), Z 1 Z pffiffi pffiffiffi 3y nDn yx Ee ¼ 1 þ 3y e P½ nDn 4x dxp1 þ 6y 0
1
2
eyx e2x dx:
0
The result follows via integration. Before closing this section let us briefly discuss the case when mð1Þ is unknown. Often is the case mð1Þ is a function of a set of parameters and hence could be estimable. For example, if F ðx; m; yÞ ¼ expfðx=m þ yðx þ ex 1ÞÞg, this is known as the Makeham sf. Since the hazard rate hðxÞ ¼ m1 þ yð1 ex Þ, then the value of mð1Þ ¼ m which could be estimated from the data. While we may not be able to get the rates we got above, consistency or strong consistency of
^ ^ g, where mð1Þ is an estimate of mð1Þ follows directly, provided of F n ðxÞ ¼ inf t40 fF n ðx þ tÞet=mð1Þ ^ course that mð1Þ is a consistent estimate.
5. Hypotheses testing In this section, we want to test H 0 : F is exponential against H 1 : F is UBA and not exponential or H 2 : F is UBAE and not exponential. Again we assume that mð1Þ is known and hence take it to be one. Thus under H 1 : F satisfies þ yÞXF ðxÞey ; Fðx
x; yX0:
ð5:1Þ
In the spirit of the pioneering work of Hollander and Proschan (1975) done for the NBU class one can take as a measure of departure from H 0 , Z 1Z 1 y þ yÞ FðxÞe ½Fðx dFðxÞ dF ðyÞ: ð5:2Þ D¼ 0
0
The test based on D is scale in variant, thus we take mð1Þ ¼ 1. If X 1 ; . . . ; X n is a random sample from F , then D may be estimated by plugging the empirical df F n in place of F . An equivalent but much simpler approach is to weigh in (5.2) by the null df which is the exponential with mean one. Thus we take the measure Z 1Z 1 y xy þ yÞ FðxÞe ½Fðx e dx dy: ð5:3Þ d 1 ¼ 0
0
Lemma 5.1. We can write that d 1 ¼ 12ð1 EeX 2EX eX Þ:
Proof. Note that Z Z 1 1 1
u d1 ¼ uF ðuÞe du F ðxÞex dx 2 0 0
ð5:4Þ
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but, Z
1
e F ðxÞ dx ¼ E x
Z
0
Next, Z
X
ex dx ¼ 1 EeX :
ð5:5Þ
0
1
du ¼ E ue FðuÞ u
0
Z
X
ueu du ¼ 1 EeX EX eX :
ð5:6Þ
0
The result follows from (5.5) and (5.6). We thus take as a measure of departure from H 0 ; d1 ¼ 1 EeX 2EX eX . This measure is estimated by n X ^d1 ¼ 1 ð1 eX i 2X i eX i Þ: n i¼1
ð5:7Þ
pffiffiffi Note that d1 is 0 under H 0 and is positive under H 1 . We also see that nðd^ 1 d1 Þ is asymptotically normal with mean 0 and t2 ¼ Varð2X eX eX 1Þ. Under H 0 , d1 ¼ 0 pffiffiffivariance
1=2 2 2 2 and t0 ¼ 27. To do the test reject H 0 if nd^ =ð27Þ bZ a , the normal value. To assess the goodness of this procedure, we evaluate its asymptotic Pitman efficacy (APE) and compare it to the values of the APE of other tests of this problem. Since there are no other tests for this problem, we may compare the APEs with those of smaller classes such as IHR or DMRL or the larger class UBAE. The APE is defined in this case as Z 1 Z 1 x 0 x 0 e f y0 ðxÞ dx þ 2 xe f y0 ðxÞ dx t0 ; ð5:8Þ APEðd1y0 Þ ¼ 0
where
f 0y0 ðxÞ
0
¼ ðd=dyÞf y ðxÞjy!y0 . Consider the following commonly used alternatives: x2
(i) The linear failure rate: f y ðxÞ ¼ ð1 þ yxÞexy =2 x (i) The Makeham: f y ðxÞ ¼ ð1 þ yð1 ex ÞÞexyðe þx1Þ Calculating the APEs of these alternatives, we get the values 0.919 and 0.510 respectively. The corresponding values of the Hollander and Proschan (1975) test for the DMRL are 0.806 and 0.288. Hence our test, which deals with the much larger class UBA is better and also simpler. Next we discuss testing H 0 against H 2 . Proceeding as above and using the result of Willmot and Cai (2000) that X is UBAE, if and only if, F ð1Þ is UBA, we have R1 R1 Lemma 5.2. If d2 ¼ 0 0 ½F ð1Þ ðx þ yÞ F ð1Þ ðxÞey exy dx dy, then d2 ¼ 32 EeX þ EX eX 1: Proof. Note that d2 can be written as Z 1Z 1 d2 ¼ ½V ðx þ yÞ V ðxÞey exy dx dy; 0
0
ð5:9Þ
ð5:10Þ
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R1 where VðxÞ ¼ 0 F ðuÞ du; xX0. But m ¼ 1, the second term in the right-hand side of (5.10) is equal to Z Z 1 Z 1 Z x 1 1 1 x 2y e dy ¼ eu du dx ¼ EeX ; ð5:11Þ VðxÞe dx F ðxÞ 2 0 2 0 0 0 and the first term of the right-hand side of (5.10) is equal to Z 1 Z 1Z 1 xy dx dy ¼ uV ðuÞeu du ¼ 2EeX þ EX eX 1: Vðx þ yÞe 0
0
ð5:12Þ
0
The result follows for (5.11) and (5.12). We estimate d2 by n X 3 X i X i ^d2 ¼ 1 þ X ie 1 : e n i¼1 2 pffiffiffi ^ Hence nðd2 d2 Þ is asymptotically normal with pmean 0 and variance ffiffiffi 17 17 1=2 . Reject H 0 if nd^ 2 =ð108 Þ bZ a , the normal s2 ¼ Varð32 ex þ X ex 1Þ. Under H 0 ; s20 ¼ 108 variate. To assess the procedure we calculate the APEs for the above-mentioned alternatives and get the values, 0.630 and 0.385, respectively.
Acknowledgements The author is indebted to the referee for a set of detailed remarks which helped improve both content and presentation.
References Ahmad, I.A., 2001. Moments inequalities of ageing families of distributions with hypothesis testing applications. J. Statist. Plann. Inference 92, 121–132. Ahmad, I.A., Mugdadi, A.R., 2004. Further moments inequalities of life distributions with hypothesis testing applications: the IFRA, NBUC and DMRL classes. J. Statist. Plann. Inference 122, 1–12. Alzaid, A.A., 1994. Aging concepts for items of unknown age. Comm. Statist. Stochastic Models 10, 649–659. Barlow, R.E., Proschan, F., 1981. Statistical Theory of Reliability and Life Testing. Silver Spring, To Begin With. Di Crescenzo, A., 1999. Dual stochastic ordering, describing aging properties of devices of unknown age. Comm. Statist. Stochastic Models 15, 561–576. Fagiuoli, E., Pellerey, F., 1997. Moments inequalities for sums of DMRL random variables. J. Appl. Probab. 34, 525–535. Hollander, M., Proschan, F., 1972. Testing whether new is better than used. Ann. Math. Statist. 43, 1136–1146. Hollander, M., Proschan, F., 1975. Tests for mean residual life. Biometrika 62, 585–594. Karr, A.F., 1993. Probability. Springer, New York, NY. Klefsjo, B., 1983. Testing exponentiality against HNBUE. Scand. J. Statist. 10, 65–75. Massart, P., 1990. The tight constant in the Dvoretzky, Kiefer, and Wolfowitz inequality. Ann. Probab. 18, 1269–1283. Willmot, G.E., Cai, J., 2000. On classes of lifetime distributions with unknown age. Probab. Eng. Inform. Sci. 14, 473–484.
Statistics & Probability Letters 69 (2004) 333–342 www.elsevier.com/locate/stapro
Some properties of classes of life distributions with unknown age Ibrahim A. Ahmad Department of Statistics and Actuarial Science, University of Central Florida, Orlando, FL 32816-2370, USA Received 4 March 2004; received in revised form 3 April 2004 Available online 8 July 2004
Abstract Some properties of the used better than aged (UBA) and the used better than aged in expectation (UBAE) classes of life distributions are given. These properties include moment inequalities and moment generating functions behavior. In addition, nonparametric estimation and testing of the survival functions of these classes are discussed. r 2004 Elsevier B.V. All rights reserved. Keywords: Used better than aged life distributions; Moment inequalities; Moment generating function; Estimation and life testing; Consistency and asymptotic normality; Pitman’s efficacy
1. Introduction Let X X0 be a random life with distribution function ðdfÞF and a survival function ðsfÞ F ¼ 1 F. At age t, define the random residual life by X tR whose sf is given by 1 dx. The mean of F ðx þ tÞ=FðtÞ; x; tX0. Assume that X has a finite mean m ¼ EðX Þ ¼ 0 FðxÞ X t is called mean residual life (MRL) and is given by Z 1 du=F ðtÞ; tX0; FðtÞ40: mðtÞ ¼ EðX t Þ ¼ FðuÞ ð1:1Þ t
Corresponding author. Tel.: +1-407-823-2289.
E-mail address: [email protected] (I.A. Ahmad). 0167-7152/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2004.06.029
ARTICLE IN PRESS I.A. Ahmad / Statistics & Probability Letters 69 (2004) 333–342
334
Further, the hazard rate of X is defined by hðtÞ ¼
d ¼ f ðtÞ ; ln FðtÞ dt FðtÞ
tX0; F ðtÞ40;
where f ðtÞ ¼ F 0 ðtÞ is the probability density of X assuming it exists. Note that if limt!1 hðtÞ ¼ hð1Þ exists and is positive, then cf. Willmot and Cai (2000) mð1Þ ¼ lim mðtÞ ¼ t!1
1 : hð1Þ
ð1:2Þ
Two classes of life distributions were introduced by Alzaid (1994) who called them the used better than aged (UBA) and the used better than aged in expectation (UBAE) classes. Precisely we have the following definitions: Definition 1.1. The df F is said to be used better than aged (UBA) if 0omð1Þo1 and for all x; tX0, þ tÞXF ðxÞet=mð1Þ : Fðx
ð1:3Þ
Definition 1.2. The df F is said to be used better than aged in expectation (UBAE) if 0omð1Þo1 and for all tX0, mðtÞXmð1Þ:
ð1:4Þ
Alzaid (1994) showed that the UBA class is a subclass of UBAE and that if F is IHR (increasing hazard rate), then F is UBA. Similar implications between UBAE, NBUE (new is better than used in expectation) and HNBUE (harmonic new is better than used in expectation) were given by Di Crescenzo (1999). More recently, Willmot and Cai (2000) showed that the UBA class includes the DMRL class (decreasing mean residual lifetime) while the UBAE includes the DVRL (decreasing variance residual lifetime). We thus have IHR DMRL UBA UBAE [ DVRL For definitions and details of the classes IHR, NBUE and DMRL, see Barlow and Proschan (1981) while for the HNBUE, see Klefsjo (1983). The DVRL class means that VarðX t Þ is decreasing in t, as t ! 1. The present work addresses probability and inferential properties of the UBA and UBAE classes. These properties seem to have not been dealt with by authors discussing these two classes. In Section 2, we give moment inequalities of these two classes. In Section 3, we give upper bounds of the moment generating functions in the two classes that guarantee their existence and finiteness. In Section 4, we estimate FðxÞ whenever assumed to be UBA (or UBAE) and demonstrate its consistency with rate. Finally, testing in these classes is discussed in Section 5.
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2. Moments inequalities The following result gives moments inequality for the UBA class. Theorem 2.1. Let F be UBA such that for some integers r; sX0; mðrþsþ2Þ ¼ EðX rþsþ2 Þo1. If lðrÞ ¼ mðrÞ =r!; rX0, then lðrþsþ2Þ Xlðrþ1Þ ½mð1Þ sþ1 :
ð2:1Þ
Proof. Since F is UBA, we have þ tÞXF ðxÞet=mð1Þ : Fðx
ð2:2Þ
Thus for all integers r; sX0, Z 1Z 1 Z xr ts F ðx þ tÞ dx duX 0
0
1
xr F ðxÞ dx
Z
0
ts et=mð1Þ dt:
ð2:3Þ
0
Now, the right-hand side (2.3) is equal to Z 1 Z sþ1 sþ1 r ½mð1Þ s! x F ðxÞ dx ¼ ½mð1Þ s!E 0
1
X
xr dx ¼
0
mðrþ1Þ s!½mð1Þ sþ1 : rþ1
ð2:4Þ
Also, the left-hand side of (2.3) is equal to Z 1Z u Z 1 Z 1 r s rþsþ1 ðu vÞ v FðuÞ dv du ¼ ws ð1 wÞr dw F ðuÞu 0 0 0 0 Z 1 urþsþ1 F ðuÞ du ¼ bðr þ 1; s þ 1Þ 0
mðrþsþ1Þ r!s! ¼ : ðr þ s þ 1Þ ðr þ s þ 1Þ!
(2.5)
The result follows from (2.4) and (2.5). Remark 2.1. Theorem 2.1 above may be extended as follows: The definition of the UBA is equivalent to the following: let x1 ; . . . ; xkþ1 be nonnegative, then F is UBA, if and only if ! kþ1 Pkþ1 X xi XF ðx1 Þe 2 xi =mð1Þ : ð2:6Þ F i¼1
Using the same methodology, we thus have: If F is UBA then Pkþ1 lPkþ1 r þkþ1 Xlðr1 þ1Þ ½mð1Þ 2 ri þk : i¼1
i
Corollary 2.1. Let r ¼ s ¼ 0, then mð2Þ X2mð1Þ mð1Þ. Corollary 2.2. Let s ¼ 0, then mðrþ2Þ Xðr þ 2Þmðrþ1Þ mð1Þ. Corollary 2.3. Let r ¼ 0, then mðsþ2Þ Xðs þ 2Þ!mð1Þ ½mð1Þ sþ1 .
ð2:7Þ
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Next, we give analogous result for the UBAE class. Toward this end we define the equilibrium life time X~ whose sf is defined by Z 1 1 du: FðuÞ ð2:8Þ F ð1Þ ðxÞ ¼ mð1Þ x Willmot and Cai (2000) showed that F is UBAE, if and only if, F ð1Þ is UBA. Using this result, we can prove the following theorem. Theorem 2.2. Let F be UBAE such that for some integers r; sX0, mðrþsþ3Þ ¼ EðX rþsþ3 Þo1. Then lðrþsþ3Þ Xlðrþ2Þ ½mð1Þ sþ1 :
ð2:9Þ
Proof. Let m~ ðrÞ denote the rth moment of X~ . Hence m~ ðrÞ ¼ mðrþ1Þ =ðr þ 1Þmð1Þ ; note that m~ ðrÞ
Z 1 Z 1 r r1 ¼r x F ð1Þ ðxÞ dx ¼ x F ðuÞ du dx mð1Þ 0 0 x Z u Z 1 Z 1 r 1 ¼ xr1 dx du ¼ xr F ðxÞ dx F ðuÞ mð1Þ 0 m 0 ð1Þ 0 Z X mðrþ1Þ 1 E xr dx ¼ : ¼ mð1Þ ðr þ 1Þmð1Þ 0 Z
ð2:10Þ
1
r1
But, F is UBAE, then F ð1Þ is UBA and thus using Theorem 2.1, m~ ðrþsþ2Þ m~ ðrþ1Þ X ½mð1Þ sþ1 ; ðr þ s þ 2Þ! ðr þ 1Þ!
(2.11)
ð2:12Þ
since m~ 1 ð1Þ is the same as mð1Þ. The result follows in light of (2.10). Remark 2.2. The above result admits the following generalization: Pkþ1 lðPkþ1 r þkþ2Þ Xlðr1 þ2Þ ½mð1Þ 2 ri þk : i¼1
ð2:13Þ
i
We can also state some interesting special cases: (i) r ¼ s ¼ 0 : m3 X3mð2Þ mð1Þ. (ii) s ¼ 0 : mðrþ3Þ Xðr þ 3Þmð1Þ. mð2Þ ½mð1Þ sþ1 . (iii) r ¼ 0 : mðsþ3Þ Xðs þ 3Þ! 2 Remark 2.3. Several authors derived moments inequalities of different families of life distributions such as IHR, IHRA, NBU, NBUE, DMRL and HNBUE as well as studied the behavior of lmþn =ln as n ! 1 for all mX1, cf. Ahmad (2001), Ahmad and Mugdadi (2004), Fagiuoli and Pellerey (1997) and the references therein. Thus the above results contribute to this literature for the UBA and UBAE classes.
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3. Existence of moment generating functions In this section, we show that the moment generating function (mgf) of X exists and is finite for the UBA class if mð1Þ exists while in the UBAE class we need mð2Þ to exist. Actually, upper bounds of the mgf’s are given. Precisely, we have Theorem 3.1. If F is UBA and if mð1Þ o1, then for all nonnegative yaðmð1ÞÞ1 , jðyÞp1 þ
mð1Þ y ; 1 ymð1Þ
ð3:1Þ
where jðyÞ ¼ EðeyX Þ. Proof. Note that
Z
jðyÞ ¼ 1 þ y
1
eyx F ðxÞ dx;
ð3:2Þ
0
for
Z
1
1þy
e F ðxÞ dx ¼ 1 þ yE yx
0
0
eyx dx ¼ 1 þ ðEðeyX 1ÞÞ ¼ jðyÞ: 0
Since F is UBA, we have Z Z 1Z 1 eyx F ðx þ tÞ dx dtX 0
Z
1
dx eyx FðxÞ 0
Z
1
et=mð1Þ dt:
ð3:3Þ
0
The left-hand side of (3.3) is equal to Z 1
Z 1Z u 1 yðuvÞ yu e e F ðuÞ du mð1Þ F ðuÞ dv du ¼ y 0 0 0 mð1Þ 1 ¼ 2 ðjðyÞ 1Þ : y y The right-hand side of (3.3) is equal to Z 1 dx ¼ mð1Þ ðjðyÞ 1Þ: eyx FðxÞ mð1Þ y 0
(3.4)
ð3:5Þ
The result now follows from (3.4) and (3.5). Theorem 3.2. If F is UBAE and if mð2Þ o1, then for all nonnegative yaðmð1ÞÞ1 , jðyÞp1 þ
mð1Þ y þ y2 ðmð2Þ mð1Þmð1Þ Þ : 1 mð1Þy
ð3:6Þ
Proof. Since F is UBAE then F ð1Þ is UBA and if j~ ð1Þ is the mgf of F ð1Þ then we have ~ ð1Þ ðyÞp1 þR m~ ð1Þ y=1 ymð1Þ. But m~ ð1Þ ¼ mð2Þ =2mð1Þ and j ~ ð1Þ ðyÞ ¼ 1=mð1Þ yðjðyÞ 1Þ, note that j 1 ~ ð1Þ ðyÞ ¼ y 0 eyx F ð1Þ ðxÞ dx þ 1. Thus the result follows from the relation j Z 1 1 1 1 eyx F ð1Þ ðxÞ dx ¼ jðyÞ : ð3:7Þ 2 2 y mð1Þ y mð1Þ y 0
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Hence, 1 mð1Þ y
ðjðyÞ 1Þp1 þ
mð2Þ y : 2mð1Þ ð1 ymð1ÞÞ
ð3:8Þ
The result now follows. It follows from the above results that if F is UBA and if mð1Þ o1 then all moments of F exist and are finite. When F is UBAE we need mð2Þ o1.
4. Estimating UBA survival functions In this section, we address the questions of estimating F ðxÞ given that it is UBA. The case of the UBAE class will be dealt with in a sequel work. Let X 1 ; . . . ; X n denote aP random sample from a life distribution F. With no restriction on F , the empirical df F n ðxÞ ¼ 1=n ni¼1 Iðx X i Þ; IðxÞ ¼ 1 if xX0 and is 0 otherwise, is a widely used nonparametric estimate of F ðxÞ. When F is UBA, we shall modify F n ðxÞ and offer an estimate that is UBA. To this end we need the following simple result.
þ tÞet=mð1Þ . Then F ¼ 1 F is UBA. Lemma 4.1. Let F be a df and define F ðxÞ ¼ inf t40 Fðx
Proof. Note that
þ y þ tÞet=mð1Þ ¼ inf F ðx þ y þ tÞeðyþtÞ=mð1Þ ey=mð1Þ F ðx þ yÞ ¼ inf Fðx t
t40
¼ inf F ðx þ uÞe
u=mð1Þ y=mð1Þ
u4y
e
X inf F ðx þ uÞeu=mð1Þ ey=mð1Þ ¼ F ðxÞ ey=mð1Þ : u40
We then propose to estimate FðxÞ by,
F n ðxÞ ¼ inf F n ðx þ tÞet=mð1Þ : t40
ð4:1Þ
Note that F n ðxÞ is UBA and also that for computational needs one can write
F n ðxÞ ¼ min F n ðx þ X ðiÞ ÞeX ðiÞ =mð1Þ ; 1pipn
ð4:2Þ
where X ð1Þ ; . . . ; X ðnÞ are the order statistics. Note also that the estimate in (4.1) or (4.2) assumes that mð1Þ is known. Let us address this case first, then will show how to deal with some of the
cases when mð1Þ is unknown. To show the consistency of F n ðxÞ and its rate, the following lemma is essential. Lemma 4.2. Let Dn ¼ supx jF n ðxÞ F ðxÞj. Then for any xX0,
jF n ðxÞ F ðxÞjp3Dn :
ð4:3Þ
Proof. Note that
jF n ðxÞ F ðxÞjpjF n ðxÞ F n ðxÞj þ Dn :
ð4:4Þ
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But, F n ðxÞpF n ðxÞ for all x we see that
jF n ðxÞ F n ðxÞj ¼ F n ðxÞ F n ðxÞ ¼ inf ½F n ðxÞ F n ðx þ tÞet=mð1Þ t40 t=mð1Þ
¼ inf f½F n ðxÞe
F n ðx þ tÞ et=mð1Þ g
t40
¼ inf fF n ðxÞet=mð1Þ F ðxÞet=mð1Þ þ F ðxÞet=mð1Þ F n ðx þ tÞ t40
þ F ðx þ tÞ F ðx þ tÞg ðF n ðx þ tÞ F ðx þ tÞÞ ¼ inf fet=mð1Þ ðF n ðxÞ FðxÞÞ t40
t=mð1Þ þ tÞ FðxÞe Þg ðFðx
p2Dn :
(4.5)
The result follows from (4.4) and (4.5). Theorem 4.1. (i) Let rX1 be an integer, then r=2 9
r Gðr=2Þnr=2 EjF n ðxÞ FðxÞj pr 2
(ii) jF ðxÞ F ðxÞj ¼ OWPI ðn1=2 ðln ln nÞ1=2 Þ. n
r EðDrn Þ. But we easily see that using Proof. (i) Note that from Lemma 4.2, EjF n ðxÞ FðxÞjp3 Corollary 3 of Massart (1990), we have Z 1 Z 1 r r1 EDn ¼ r x PðDn 4xÞ dx ¼ r xr1 Pðn1=2 Dn 4n1=2 xÞ dx 0 0 Z 1 pffiffiffi r=2 ur1 Pð nDn uÞ du ¼ rn 0
p2rnr=2
Z 0
1
2
ur1 e2u du ¼
rnr=2 2r=2
Gðr=2Þ:
(4.6)
The result is now immediate. (ii) Follows directly from Lemma 4.2 and the standard law of iterated logarithm of Dn , cf. Karr (1993, p. 207). In the following result, we give a rate of convergence in the moment generating function of
jF n ðxÞ F ðxÞj. Theorem 4.4. For any y40, pffiffi pffiffiffiffiffiffi 2 Eey njF n ðxÞF ðxÞj p1 þ 3 2pye18y :
Proof. Using Lemma 4.2, we get that pffiffi pffiffi Eey njF n ðxÞF ðxÞj pEe3y nDn :
ð4:7Þ
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Note that using Corollary 3 of Massart (1990), Z 1 Z pffiffi pffiffiffi 3y nDn yx Ee ¼ 1 þ 3y e P½ nDn 4x dxp1 þ 6y 0
1
2
eyx e2x dx:
0
The result follows via integration. Before closing this section let us briefly discuss the case when mð1Þ is unknown. Often is the case mð1Þ is a function of a set of parameters and hence could be estimable. For example, if F ðx; m; yÞ ¼ expfðx=m þ yðx þ ex 1ÞÞg, this is known as the Makeham sf. Since the hazard rate hðxÞ ¼ m1 þ yð1 ex Þ, then the value of mð1Þ ¼ m which could be estimated from the data. While we may not be able to get the rates we got above, consistency or strong consistency of
^ ^ g, where mð1Þ is an estimate of mð1Þ follows directly, provided of F n ðxÞ ¼ inf t40 fF n ðx þ tÞet=mð1Þ ^ course that mð1Þ is a consistent estimate.
5. Hypotheses testing In this section, we want to test H 0 : F is exponential against H 1 : F is UBA and not exponential or H 2 : F is UBAE and not exponential. Again we assume that mð1Þ is known and hence take it to be one. Thus under H 1 : F satisfies þ yÞXF ðxÞey ; Fðx
x; yX0:
ð5:1Þ
In the spirit of the pioneering work of Hollander and Proschan (1975) done for the NBU class one can take as a measure of departure from H 0 , Z 1Z 1 y þ yÞ FðxÞe ½Fðx dFðxÞ dF ðyÞ: ð5:2Þ D¼ 0
0
The test based on D is scale in variant, thus we take mð1Þ ¼ 1. If X 1 ; . . . ; X n is a random sample from F , then D may be estimated by plugging the empirical df F n in place of F . An equivalent but much simpler approach is to weigh in (5.2) by the null df which is the exponential with mean one. Thus we take the measure Z 1Z 1 y xy þ yÞ FðxÞe ½Fðx e dx dy: ð5:3Þ d 1 ¼ 0
0
Lemma 5.1. We can write that d 1 ¼ 12ð1 EeX 2EX eX Þ:
Proof. Note that Z Z 1 1 1
u d1 ¼ uF ðuÞe du F ðxÞex dx 2 0 0
ð5:4Þ
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341
but, Z
1
e F ðxÞ dx ¼ E x
Z
0
Next, Z
X
ex dx ¼ 1 EeX :
ð5:5Þ
0
1
du ¼ E ue FðuÞ u
0
Z
X
ueu du ¼ 1 EeX EX eX :
ð5:6Þ
0
The result follows from (5.5) and (5.6). We thus take as a measure of departure from H 0 ; d1 ¼ 1 EeX 2EX eX . This measure is estimated by n X ^d1 ¼ 1 ð1 eX i 2X i eX i Þ: n i¼1
ð5:7Þ
pffiffiffi Note that d1 is 0 under H 0 and is positive under H 1 . We also see that nðd^ 1 d1 Þ is asymptotically normal with mean 0 and t2 ¼ Varð2X eX eX 1Þ. Under H 0 , d1 ¼ 0 pffiffiffivariance
1=2 2 2 2 and t0 ¼ 27. To do the test reject H 0 if nd^ =ð27Þ bZ a , the normal value. To assess the goodness of this procedure, we evaluate its asymptotic Pitman efficacy (APE) and compare it to the values of the APE of other tests of this problem. Since there are no other tests for this problem, we may compare the APEs with those of smaller classes such as IHR or DMRL or the larger class UBAE. The APE is defined in this case as Z 1 Z 1 x 0 x 0 e f y0 ðxÞ dx þ 2 xe f y0 ðxÞ dx t0 ; ð5:8Þ APEðd1y0 Þ ¼ 0
where
f 0y0 ðxÞ
0
¼ ðd=dyÞf y ðxÞjy!y0 . Consider the following commonly used alternatives: x2
(i) The linear failure rate: f y ðxÞ ¼ ð1 þ yxÞexy =2 x (i) The Makeham: f y ðxÞ ¼ ð1 þ yð1 ex ÞÞexyðe þx1Þ Calculating the APEs of these alternatives, we get the values 0.919 and 0.510 respectively. The corresponding values of the Hollander and Proschan (1975) test for the DMRL are 0.806 and 0.288. Hence our test, which deals with the much larger class UBA is better and also simpler. Next we discuss testing H 0 against H 2 . Proceeding as above and using the result of Willmot and Cai (2000) that X is UBAE, if and only if, F ð1Þ is UBA, we have R1 R1 Lemma 5.2. If d2 ¼ 0 0 ½F ð1Þ ðx þ yÞ F ð1Þ ðxÞey exy dx dy, then d2 ¼ 32 EeX þ EX eX 1: Proof. Note that d2 can be written as Z 1Z 1 d2 ¼ ½V ðx þ yÞ V ðxÞey exy dx dy; 0
0
ð5:9Þ
ð5:10Þ
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R1 where VðxÞ ¼ 0 F ðuÞ du; xX0. But m ¼ 1, the second term in the right-hand side of (5.10) is equal to Z Z 1 Z 1 Z x 1 1 1 x 2y e dy ¼ eu du dx ¼ EeX ; ð5:11Þ VðxÞe dx F ðxÞ 2 0 2 0 0 0 and the first term of the right-hand side of (5.10) is equal to Z 1 Z 1Z 1 xy dx dy ¼ uV ðuÞeu du ¼ 2EeX þ EX eX 1: Vðx þ yÞe 0
0
ð5:12Þ
0
The result follows for (5.11) and (5.12). We estimate d2 by n X 3 X i X i ^d2 ¼ 1 þ X ie 1 : e n i¼1 2 pffiffiffi ^ Hence nðd2 d2 Þ is asymptotically normal with pmean 0 and variance ffiffiffi 17 17 1=2 . Reject H 0 if nd^ 2 =ð108 Þ bZ a , the normal s2 ¼ Varð32 ex þ X ex 1Þ. Under H 0 ; s20 ¼ 108 variate. To assess the procedure we calculate the APEs for the above-mentioned alternatives and get the values, 0.630 and 0.385, respectively.
Acknowledgements The author is indebted to the referee for a set of detailed remarks which helped improve both content and presentation.
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