Maximum likelihood estimators under progressive Type-I interval censoring

Statistics and Probability Letters xx (xxxx) xxx–xxx

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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Q1

Q2

Maximum likelihood estimators under progressive Type-I interval censoring Sonal Budhiraja a,∗ , Biswabrata Pradhan a , Debasis Sengupta b a

SQC and OR Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, PIN—700108, India

b

Applied Statistics Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, PIN—700108, India

article

info

Article history: Received 5 July 2016 Received in revised form 16 December 2016 Accepted 16 December 2016 Available online xxxx

abstract The consistency and asymptotic normality of the maximum likelihood estimators (MLEs), based on progressively type-I interval censored (PIC-I) data are proved under appropriate regularity conditions. The information obtained in the PIC-I setup is compared with that of grouped data and also with progressively type-I censored (PC-I) data. © 2016 Elsevier B.V. All rights reserved.

Keywords: Asymptotic normality Consistency Interval censoring Progressive type-I censoring Regularity conditions

1. Introduction

1

The PIC-I scheme, introduced by Aggarwala (2001), can be described as follows. Suppose n identical items are placed on a life test at time T0 = 0. The items on test are inspected at pre-specified times T1 < T2 < · · · < Tk , where Tk is the scheduled termination time of the experiment and k is pre-fixed. At the jth inspection time Tj , Rj surviving items are randomly removed from the life test, for j = 1, . . . , k − 1. The experiment terminates at Tk with Rk being the number of surviving items at that time. In this censoring scheme, the lifetimes of the items are unobserved. However, the number of failures occurring in different inspection intervals and the number of randomly removed items at pre-specified inspection times are observed. Let Dj be the number of failures in the interval (Tj−1 , Tj ] and Sj , the number of surviving items at Tj . Note that Sj is a random variable and Rj should not be greater  than  Sj . In this work, we consider that for given pre-specified proportion, p1 , . . . , pk−1 , Rj ’s are determined by Rj = Sj pj , for j = 1, 2, . . . , k − 1, where ⌊x⌋ is the greatest integer less than or equal to x. If Rj = 0, for j = 1, 2, . . . , k − 1, then PIC-I scheme reduces to grouped data. Suppose Hj denotes the history up to the inspection time Tj , where Hj = {D1 , R1 , . . . , Dj , Rj }, for j = 1, . . . , k and H0 is empty. Then Hk represents the data arising from PIC-I censoring scheme. Let Nj be the number of items at risk at the beginning of the jth interval (Tj−1 , Tj ], for j = 1, 2, . . . , k. Then, we have N1 = n and Nj = Nj−1 − Dj−1 − Rj−1 , for j = 2, 3, . . . , k. Also, we have n = j=1 Dj + Rj . Suppose X denotes the lifetime with distribution function F (·; θ) and probability density function f (·; θ), where θ ∈ Θ , an open subset of Rm and m is the number of parameters. Let q = (q1 , q2 , . . . , qk ), where qj is the conditional probability that an item which is at risk at time Tj−1 , will fail by time Tj , for j = 1, 2, . . . , k, where T0 = 0. Then qj is given by

k 

qj =

∗

F (Tj ;θ)−F (Tj−1 ;θ) 1−F (Tj−1 ;θ)



, for j = 1, 2, . . . , k. Note that Dj |Hj−1 ∼ Binom(Nj , qj ), for j = 1, 2, . . . , k. Then, it easily follows

Corresponding author. E-mail address: [email protected] (S. Budhiraja).

http://dx.doi.org/10.1016/j.spl.2016.12.013 0167-7152/© 2016 Elsevier B.V. All rights reserved.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

2

1

2

S. Budhiraja et al. / Statistics and Probability Letters xx (xxxx) xxx–xxx

(see Wu et al., 2008) that E[Nj ] = n

j −1 

(1 − pl )(1 − ql ) = n

l =1

j −1 

  (1 − pl ) 1 − F (Tj−1 ; θ) ,

(1)

l =1

16

There exists a number of works on inference for different lifetime distributions with PIC-I data. See for example, Aggarwala (2001), Ng and Wang (2009), Chen and Lio (2010). They have derived the MLEs of the model parameters and applied the asymptotic properties of the MLEs for inference. Asymptotic properties are also used for determination of life testing plans under PIC-I scheme. See for example, Yang and Tse (2005), Wu et al. (2008), Huang and Wu (2008), Lin et al. (2009), Wu and Huang (2010), Lin et al. (2011) and Kus et al. (2013). To the best of our knowledge, there does not exist any work establishing the asymptotic properties of the MLEs. In this work, we establish the consistency and asymptotic normality of the MLEs of θ with PIC-I data under appropriate regularity conditions. Note that the unobserved lifetimes of the items under test are independent and identically distributed (i.i.d.) with common distribution F (.; θ). However, the data observed under PIC-I scheme are not independent. Let nj be the observed number of items at risk at the beginning of the jth interval, indexed by the labels 1, 2, . . . , nj , and let δlj be the indicator function which takes value 1, if the lth surviving item at the beginning of the jth interval, fails in the jth interval, and nj nj 0 otherwise, for l = 1, 2, . . . , nj , j = 1, 2, . . . , k. Note that l=1 δlj = dj and l=1 (1 − δlj ) = nj − dj , where dj is the observed value of Dj for j = 1, 2, . . . , k. Since the items surviving at the beginning of the jth interval have identical histories of survival and non-withdrawal, they may be regarded as i.i.d. Thus, the likelihood function can be written as

17

L(Hk |θ) ∝

3 4 5 6 7 8 9 10 11 12 13 14 15

18

k nj

l (θ) =

j =1

l=1

nj k   

δlj

qj (1 − qj )1−δlj . The log likelihood function, after ignoring the constant of proportionality, is given by

 δlj log qj + (1 − δlj ) log(1 − qj ) .

(2)

j =1 l =1

23

Since the summands of the log-likelihood are not i.i.d., the asymptotic properties do not follow readily from standard theorems. Some adjustment is needed for deriving the requisite results. The rest of the paper is organized as follows. The asymptotic properties of the MLEs are proved under certain regularity conditions in Section 2. The information from PIC-I data is compared with that from PC-I and grouped data in Section 3. A simulation study is undertaken to assess the performance of the MLEs in Section 4.

24

2. Asymptotic properties of MLEs

19 20 21 22

26

Let θˆ be the MLE of θ . We consider the following regularity conditions to establish the consistency and asymptotic normality of θˆ .

27

Regularity conditions:

25

28 29 30 31 32 33 34 35 36

37 38 39

40

41 42 43 44 45 46

(I) The unobserved lifetimes X1 , X2 , . . . , Xn are i.i.d. with common distribution function F (.; θ) and density function f (.; θ). (II) The support of f (.; θ) is independent of θ . (III) The parameter space Θ contains an open set Θ0 of which the true parameter θ 0 is an interior point. ∂ 3 F (x;θ)

(IV) For almost all x, the distribution function F (x; θ) admit all the third derivatives ∂θ ∂θ ∂θ for all θ ∈ Θ0 and u, v, w = u v w 1, 2, . . . , m. Also all the first, second and third order derivatives of F (x; θ) with respect to the parameters are bounded for all θ ∈ Θ0 . (V) The Tj ’s are chosen in such a way that (a) 0 < qj < 1 for j = 1, 2, . . . , k.   (b) ∇θ q is a matrix of rank m, where ∇θ q =

∂ qj ∂θu

m×k

for j = 1, 2, . . . , k and u = 1, 2, . . . , m.

Remark. The condition V(b) ensures that there is no function of θ with respect to which the derivatives of all the qj ’s is zero. This requirement is similar to the notion of identifiability, but not quite the same. In the present set-up, identifiability amounts to the condition, for any θ 1 , θ 2 ∈ Θ0 ,

such that θ 1 ̸= θ 2 =⇒ q(θ 1 ) ̸= q(θ 2 ).

(3)

(i) When m = k = 2, the condition V(b) implies (3), that is non-identifiability implies that the gradient matrix ∇θ q is rank deficient. We consider two families of intersecting curves. Suppose there are θ 1 , θ 2 ∈ Θ0 such that θ 1 ̸= θ 2 but qj (θ 1 ) = qj (θ 2 ) ∂q

for j = 1, 2. Consider the set {θ ∈ Θ0 |q1 (θ) = q1 (θ 1 )} and a path within this set connecting θ 1 and θ 2 . Note that ∂θ1 is orthogonal to the path at any point along the path. Since q2 (θ 1 ) = q2 (θ 2 ), Rolle’s theorem ensures that there is a point ∂q on the path where ∂θ2 is also orthogonal to the path.

S. Budhiraja et al. / Statistics and Probability Letters xx (xxxx) xxx–xxx

(ii) However, conditionV(b) does not generally imply identifiability. Let q1 (θ) = 1 −

and q4 (θ) =

2 5



7 2

3

1







(θ + θ ), q2 (θ) = 1 − (1 − θ1 ) + (1 − θ2 ) , q3 (θ) = − θ + θ + 5θ + 2θ1 + 1 ,   225 3 2 2 − 19 θ1 + θ1 + 10θ2 − 2θ1 − 2θ2 + 1 such that θ = (θ1 , θ2 ) ∈ (0.3, 0.7) × (0.3, 0.7). Consider 2 1



2 2

2

10 11

2

5 2

50 3 19 1

2 1

2 2

θ = (0.4, 0.6) and θ = (0.6, 0.4). It may be verified that θ ̸= θ but qj (θ ) = qj (θ ) for j = 1, 2, . . . , k, that is the parameters are non-identifiable but the gradient matrix ∇θ q is full rank over (0.3, 0.7) × (0.3, 0.7). 1

2

1

2

1

2

(iii) The reverse implication does not hold. Let q1 (θ) = 0.5 + 4(θ1 − 0.5)3 and q2 (θ) = θ2 , where θ = (θ1 , θ2 ) ∈ (0, 1) × (0, 1). The parameters are identifiable as qj (θ 1 ) = qj (θ 2 ) =⇒ θ 1 = θ 2 yet the gradient matrix ∇θ q is singular at θ = (0.5, θ2 ) for all θ2 ∈ (0, 1). Hence identifiability does not imply that the gradient matrix is rank deficient.

2

3 4 5 6 7 8 9

Now, we consider the following lemma and theorem on asymptotic results of the MLEs.

10

Lemma 1. Under the regularity conditions (I)–(IV), the first and second derivatives of θ satisfy the following equations

11

(i) Eθ (ii) Eθ

 

∂ l (θ) ∂θu



= 0, for u = 1, 2, . . . , m.   2  ∂ l (θ) ∂ l(θ) ∂ l(θ) = E − ∂θ , for u, v = 1, 2, . . . , m. . θ ∂θu ∂θv u ∂θv

12

13

By using Lemma 1, we deduce that the Fisher information matrix for θ is given by I(θ) =



Iuv (θ)



, where Iuv (θ) =

14

 2  ∂ l (θ) E − ∂θ ∂θ , for u, v = 1, 2, . . . , m. u v

15

Theorem 1. If the regularity conditions (I)–(V) hold, then

16

(a) θˆ is consistent for θ and

17

(b) θˆ is asymptotically normal with mean θ 0 and variance–covariance matrix [I(θ 0 )]−1 .

18

Proof of Lemma 1. It follows from (2) that

19

 nj  nj k  k   δlj ∂ l (θ)  1 − δlj ∂ qj Ulj,u , (4) = − = ∂θu qj 1 − qj ∂θu j =1 l =1 j =1 l =1   δ 1−δ ∂q where Ulj,u = qlj − 1−qlj ∂θ j . j j   N  u  k j ∂ l (θ) = E E U | N . Note that E(δlj |nj ) = qj , for l = 1, 2, . . . , nj and By using (4), we have E ∂θ lj , u j l =1 j =1 u j = 1, 2, . . . , k. This implies      qj 1 − qj ∂ qj  E Ulj,u nj = − = 0. (5) qj 1 − qj ∂θu   ∂ l (θ) Hence E ∂θ = 0. u k k nj nj′ ∂ l (θ) ∂ l (θ) Next we prove part (ii) of Lemma 1. By using (4), we have ∂θ ∂θ = U U ′ ′ . Suppose j < j′ . ′ u v    j=1 j =1 l=1 l′ =1 ′ lj,u l j ,v  Then for given nj′ , nj is fixed and hence E Ulj,u Ul′ j′ ,v |nj′ = Ulj,u E Ul′ j′ ,v |nj′ = 0. Similarly, when j < j, E Ulj,u Ul′ j′ ,v |nj =        δlj δ ′ δlj (1−δ ′ ) (1−δlj )δ ′ (1−δlj )(1−δ ′ )  Ul′ j′ ,v E Ulj,u |nj = 0. Also, we have E Ulj,u Ul′ j,v |nj = E q2l j − qj (1−qlj )j − qj (1−qjl)j + (1−q )2 l j nj = 0 and j j  2     δlj 1−δlj ∂ qj ∂ qj ∂ qj ∂ qj 1 E Ulj,u Ulj,v |nj = E − 1−qj nj ∂θu ∂θv = qj (1−qj ) ∂θu ∂θv . Thus, qj   k ∂ l (θ) ∂ l (θ) E[Nj ] ∂ qj ∂ qj = . ∂θu ∂θv q ( 1 − qj ) ∂θu ∂θv j j =1  2  ∂ l (θ) Next, we derive the expression for E − ∂θ ∂θ . We have u v

20

21

22 23

24

25

26 27

28

29



E

n

j k   ∂ 2 l (θ) = ∂θu ∂θv j =1 l =1



δlj qj

−

1 − δlj 1 − qj



∂ 2 qj − ∂θu ∂θv



δlj q2j

(6)

30

31

+

1 − δlj

(1 − qj )2



∂ qj ∂ qj ∂θu ∂θv

 .

32

4

1

2

3

4 5

6 7

S. Budhiraja et al. / Statistics and Probability Letters xx (xxxx) xxx–xxx

Now, E



δlj

−

qj

1−δlj



1−qj

∂ 2 qj ∂θu ∂θv



 = 0 and E

δlj q2j

+

1−δlj



(1−qj )2

∂ qj ∂ qj ∂θu ∂θv

 =

∂ qj ∂ qj 1 qj (1−qj ) ∂θu ∂θv

. Thus

    2 Nj k  k   1 ∂ qj ∂ qj E[Nj ] ∂ qj ∂ qj ∂ l (θ) =E = . E − ∂θu ∂θv qj (1 − qj ) ∂θu ∂θv qj (1 − qj ) ∂θu ∂θv j =1 l =1 j=1

(7)

By using Eqs. (6) and (7) the desired result is proved. Before proving Theorem 1, we state and prove the following lemmas. For the sake of convenience, we write F (Tj ) in place of F (Tj ; θ) in all subsequent expressions. Lemma 2. Under the regularity conditions (I)–(IV), the ratio j = 1, 2, . . . , k.

Nj n

converges in probability to a finite number bj , as n → ∞, for

10

   Nj −1 ( 1 − p )( 1 − q ) . Thus, E = jl= l l l =1 1 (1 − pl )(1 − ql ) = bj , say. It is clear n that bj < ∞. Now, it is enough to show that the variance of Nj /n tends to zero as n → ∞. We have        j −1 l 2 Nj Nj 1  Nj (1 − pl′ )(1 − ql′ )(1 − pl )ql . =E − E2 = Var

11

Thus, Var

12

Lemma 3. Under the regularity conditions (I)–(V), each element of the matrix I(θ) is finite for all θ in Θ0 .

8 9

Proof of Lemma 2. By (1), we have E[Nj ] = n

n

  Nj n

n

n

j−1

n l =1 ′ l =1

→ 0, when n → ∞. Hence the proof.

Proof of Lemma 3. By using (6), we have Iuv (θ) = 1 1−F (Tj−1 )



∂ F (Tj ) ∂θu

− (1 −

ζ = max sup 1≤j≤k θ∈Θ

0

∂ F (T ) qj ) ∂θj−1 u

1 1 − F (Tj )

14

E[Nj ] ∂ qj ∂ qj j=1 qj (1−qj ) ∂θu ∂θv

k

(8) for u = 1, 2, . . . , m.

(9)

Then, by using (8) and (9), we have

   ∂ qj     ∂θ  ≤ 2ζ Au = Bu , say. u

(10)

Note that for fixed n, E[Nj ] < n. Also, 0 <

16

Lemma 4. Under the regularity conditions (I)–(V), the information matrix is positive definite for all θ in Θ0 .

18 19

20

21 22 23

24 25 26

=

, for u = 1, 2, . . . , m. We define the following bounds

15

17

∂q

, for u, v = 1, 2, . . . , m. Note that ∂θ j u

.

   ∂ F (Tj )   , Au = max sup  1≤j≤k θ∈Θ ∂θu  0 13



1 qj (1−qj )

< ∞, ∀θ ∈ Θ0 , since 0 < qj < 1. Thus, Iuv (θ) < ∞. Hence the proof.

Proof of Lemma 4. As I(θ) is the variance–covariance matrix of the score vector, it is non-negative definite. We prove that it is a positive definite matrix by contradiction. Let us assume that I(θ) is a singular matrix for some θ ∈ Θ0 . Then for a fixed θ there exists a vector a(θ) ̸= 0 such that a(θ)T I(θ)a(θ) =

k 

E[Nj ]

j =1

qj (1 − qj )

a(θ)T ∇θ qj



2

= 0.

(11)

Since 0 < qj < 1 and E[Nj ] > 0, for j = 1, 2, . . . , k, Eq. (11) holds if a(θ)T ∇θ qj = 0 for all j = 1, 2, . . . , k at θ . Thus, ∇θ q is rank deficient, which contradicts the regularity condition V(b). So we can conclude that under the given set up, I(θ) is a non-singular matrix and a(θ)T I(θ)a(θ) > 0. Hence I(θ) is a positive definite matrix for all θ ∈ Θ0 . Lemma 5. Suppose the regularity conditions (I)–(V) hold. Let Θ0 be an open subset in the parameter space Ω containing  3 the  ∂ l (θ)  ∂ 3 l (θ) 0 true parameter θ . Then, all the third derivatives ∂θ ∂θ ∂θ exist and there also exists a bound Guvw (Hk ), such that  ∂θ ∂θ θ  ≤ u

v

w

Guvw (Hk ) on Θ0 with E[Guvw (Hk )] = nguvw , where guvw is finite for u, v, w = 1, 2, . . . , m.

u

v w

S. Budhiraja et al. / Statistics and Probability Letters xx (xxxx) xxx–xxx

5

Proof of Lemma 5. We have

∂ 3 l (θ) = ∂θu ∂θv ∂θw

1

nj

k

δlj

  

−

1 − δlj



∂ 3 qj ∂θu ∂θv ∂θw

2

(1 − qj )    δlj ∂ 2 qj ∂ qj ∂ 2 qj ∂ qj ∂ 2 qj ∂ qj 1 − δlj − + + + (1 − qj )2 ∂θu ∂θv ∂θw ∂θv ∂θw ∂θu ∂θw ∂θu ∂θv q2j    δlj 1 − δlj ∂ qj ∂ qj ∂ qj +2 3 − . 3 (1 − qj ) ∂θu ∂θv ∂θw qj qj

j =1 l =1

∂2q

3

(12)

4

∂3q

To obtain the required bound, we need to derive the bounds for ∂θ ∂θj and ∂θ ∂θ j∂θ for j = 1, 2, . . . , k. For u, v, w = u v u v w 1, 2, . . . , m, we define the following bounds

 ∂ 2 F (Tj )  . Auv = max sup  1≤j≤k θ∈Θ ∂θu ∂θv  0 



(13)

 ∂ 3 F (Tj )  . Auvw = max sup  1≤j≤k θ∈Θ ∂θu ∂θv ∂θw  0 



(14)

Considering the second derivatives of qj ’s with respect to parameters and by using (8)–(10) and (13), for u, v = 1, 2, . . . , m, we get,

 2  ∂ qj   ∂θ ∂θ u

v

   ≤ 2ζ 2 Au Av + ζ (2Auv + Bv Au ) = Buv , say. 

(15)

Similarly by taking the third derivatives of qj ’s with respect to θ and using (8)–(10), (13) and (14), we get (See Budhiraja et al., 2016)

    ∂ 3 qj   3 2    ∂θ ∂θ ∂θ  ≤ 4ζ Au Av Aw + ζ 2Avw Au + Av (2Auw + Bw Au ) + Aw (2Auv + Bv Au ) u v w   + ζ 2Auvw + Bvw Au + Bv Auw + Bw Auv = Buvw , say.

(16)

Now we prove Theorem 1 by using Lemmas 2–5.

(ii)

p

→ 0,

n ∂θu 1 ∂ 2 l (θ) n ∂θu ∂θv

p

→ −Juv ,

as n → ∞, where Juv =

k  j =1

∂ l (θ)

= n

p

→ 0, as n → ∞. So by using Lemma 2, it follows that Proof of part (ii): By using (6), we have δlj q2j

+

1−δlj (1−qj )2

∂ qj ∂ qj ∂θu ∂θv

9

11

12

13 14

16 17

nj j =1 n

k

(17)

18

Ulj,u . Note that for given nj , Ulj,u ’s are i.i.d., for

19

∂ qj ∂ qj . qj (1 − qj ) ∂θu ∂θv bj

 n j 1



l =1

nj

l = 1, . . . , nj , with E[Ulj,u |nj ] = 0 by (5). Now, by weak law of large numbers (WLLN), for given nj , we get



8

as n → ∞.

Proof of part (i): By using (4), we have ∂θ u



7

15

Proof of Theorem 1(a). The proof of consistency of θˆ follows from Lehmann and Casella (1998, Theorem 5.1, pp. 463) in view of Lemma 5 and the following results. 1 ∂ l (θ)

6

10

  Further define γ1 = max1≤j≤k supθ∈Θ0 q1 and γ2 = max1≤j≤k supθ∈Θ0 1−1q . Let k = γ1 γ2 Buvw + 2 γ13 + γ23 Bu Bv Bw + (γ12 + j j     3   n   ≤ kj=1 l=j 1 k, say. γ22 ) Buv Bw + Bvw Bu + Bwu Bv . Then by using the bounds (10), (15) and (16) in (12), we get,  ∂θ∂u ∂θlv(θ) ∂θw  k k Now E[Guvw (Hk )] = nk j=1 bj , where guvw = k j=1 bj < ∞. Hence the proof.

(i)

5

1 ∂ 2 l (θ) n ∂θu ∂θv

k

nj

j=1 n

=

 n j 1

nj j=1 n

k

l =1

nj



Ulj,u → 0 as n → ∞. Hence

 n j 1 nj

p

l =1



∂q ∂q

with E Vlj,uv |nj = − q (11−q ) ∂θ j ∂θ j . By WLLN, for given nj , we have u v j j



Hence by using Lemma 2, we get



1 ∂ 2 l (θ) n ∂θu ∂θv

p

→−

bj ∂ qj ∂ qj j=1 qj (1−qj ) ∂θu ∂θv

k

=

Vlj,uv , where Vlj,uv

= −Juv .

1 nj

nj

l =1

1 ∂ l (θ) n ∂θu



δlj qj

1 nj

nj

l =1

Ulj,u

→ 0, as n → ∞.  2 1−δ ∂ q − 1−qljj ∂θu ∂θj v −

p

20

p

∂q ∂q

Vlj,uv → − q (11−q ) ∂θ j ∂θ j . u v j j

21

22

23

24

6

1

2

3

S. Budhiraja et al. / Statistics and Probability Letters xx (xxxx) xxx–xxx

Proof of Theorem 1(b). Consider the Taylor’s Series expansion of ∂θ∂ l (θ), for u = 1, 2, . . . , m. u

∂ l (θ) ∂ l (θ)|θ=θˆ = | 0 ∂θu ∂θu θ=θ

where θ ∗ is a point on line segment connecting θˆ and θ 0 . Then,

 4

5

ˆ = ∇θ l (θ 0 ) + ∇θ l (θ)

7

8

9

m 1

0

2

0

 √ ∂ 3 1 − ∇θ l (θ ) − ∇θ l (θ)|θ=θ∗ (θˆw − θw ) n(θˆ − θ 0 ) = √ ∇θ l (θ 0 ). n n w=1 ∂θw n 1

m 1

0

2

0

By Lemma 5 and the consistency of θˆ , we have 1

m 1

n

n w=1

− ∇θ2 l (θ 0 ) −

(θˆw − θw0 )

1 n



√ ∇θ l (θ ) = 0

n

k 



j =1

nj n

p ˆ − θw0 ) ∂ ∇ 3 l (θ)|θ=θ∗ → 0. Then, by using (17), it follows that ∂θw θ

∂ 3 p ∇θ l (θ)|θ=θ∗ → J (θ 0 ). ∂θw

k 

···

Sj1

(18)

w=1 (θw

m

(19)

Next, we find the limiting distribution of √1n ∇θ l (θ 0 ). Define Sju = √1n 1

10

 ∂ 3 ∇ l (θ)|θ=θ∗ n(θˆ − θ 0 ). ∇θ l (θ ) + (θˆw − θw ) n n w=1 ∂θw θ

1

ˆ = 0, the above expression can be represented as Since ∇θ l(θ) 

6

m m  m   ∂ 2 l (θ) ∂ 3 l (θ) + (θˆv − θv0 ) |θ=θ0 + (θˆ v − θv0 )(θˆw − θw0 ) |θ=θ∗ , ∂θu ∂θv ∂θu ∂θv ∂θw v=1 v=1 w=1



j=1





=

nj n

nj

l =1

j

Ulj,u . Then, we have

T .

Sjm

1 qj (1−qj )



(20)

∂ qj ∂θu

2

= σj2,u , say. Then E[Sju ] = 0 and Var(Sju ) = σj2,u , for

11

Note that E[Ulj,u |nj ] = 0 and Var Ulj,u |nj

12

j = 1, 2, . . . , k and u = 1, 2, . . . , m. Now by central limit theorem, for given nj , Ulj,u ∼ N (0, σj2,u ), for j = 1, 2, . . . , k.

a

13 14

Further, define S j = (Sj1 , Sj2 , . . . , Sjm )T and the variance–covariance matrix of S j as Σj = j = 1, 2, . . . , k.



Cov(Sju , Sjv )



m×m

for

a

15 16 17

Then, S j |nj ∼ Nm (0, Σj ), for j = 1, . . . , k. Further, note that the conditional distribution of S j given nj is independent of nj , for j = 1, . . . , k. Also, Cov(Sju , Sj′ v ) = 0, for j ̸= j′ = 1, 2, . . . , k. Hence S 1 , . . . , S k are uncorrelated. As a result, the



distribution of S = S 1

S2

...

Sk

T

is asymptotically Nkm (0, Σ ), where Σ = diag (Σ1 , Σ2 . . . , Σk ).

z 18

19

Let Z be a matrix of order m × km defined as Z = the uth element is



nj n

z21

z12 z22

zm1

zm2

11

...

... ... ... ...

z1k z2k

 where zuj is a row vector of length m such that

...

zmk

and 0 otherwise, for u = 1, 2, . . . , m and j = 1, 2, . . . , k. Then, by (20), we have √1n ∇θ l (θ 0 ) = ZS .

a

20

Since S ∼ Nkm (0, Σ ), it follows that a

21

ZS ∼ Nm (0, ZΣ ZT ).

(21) nj j=1 n Co

22

The uv th element of ZΣ ZT is given by (ZΣ ZT )uv =

23

Sjv ) = Juv (θ ). By using (18), (20) and (21), it follows that

25

nj j=1 n Cov

k

p

(Sju , Sjv ) →

k

j =1

bj Cov(Sju ,

0

1

24

v(Sju , Sjv ). Note that

k

a

√ ∇θ l (θ 0 ) ∼ Nm (0, J (θ 0 )).

(22)

n

Finally, by using (19) and (22), we have

√

a

 

n(θˆ − θ 0 ) ∼ N 0, J (θ 0 )

−1 

. Hence θˆ is asymptotically normal with mean θ 0

27

and variance–covariance matrix [I(θ 0 )]−1 , where I(θ 0 ) = nJ (θ 0 ). Note that I(θ) is finite and invertible by Lemmas 3 and 4, respectively. Thus, the part (b) of Theorem 1 is proved.

28

3. Comparison of loss of information in different censoring schemes

26

29 30

Note that PIC-I reduces to simple grouping of data when p1 = p2 = · · · = pk−1 = 0. While the censoring is generally done to reduce cost, it might lead to loss of information. We now show that this is indeed the case.

S. Budhiraja et al. / Statistics and Probability Letters xx (xxxx) xxx–xxx

7

Table 1 The AE, MSE, CP of asymptotic 95% confidence intervals and ratio of MSE and sample variance for different sample sizes when (α, λ) = (1.5, 1).

α

n

15 30 50 75 100

λ

AE

MSE

CP

MSE /S 2

AE

MSE

CP

MSE /S 2

1.6512 1.5661 1.5319 1.5244 1.5194

0.2521 0.0943 0.0485 0.0319 0.0236

0.9481 0.9476 0.9493 0.9501 0.9534

1.0996 1.0485 1.0213 1.0189 1.0161

1.0344 1.0188 1.0095 1.0071 1.0055

0.0486 0.0219 0.0131 0.0085 0.0065

0.9229 0.9429 0.9455 0.9476 0.9463

1.0250 1.0163 1.0068 1.0058 1.0046

Result 1. In a grouped data set-up the Fisher information of parameters underlying the group probabilities is reduced if a certain fraction of items are progressively censored at the end of grouping intervals. Proof. By using Eqs. (1) and (6), for PIC-I, the (u, v)th element of the Fisher information matrix is given by Iuv (θ) =  n

j−1 (1−pl )(1−ql ) l=1

k

qj (1−qj )

j =1

∂ qj ∂ qj ∂θu ∂θv

for u, v

= 1, 2, . . . , m. In case p1

= p2

= · · · = pk−1

= 0, PIC-I scheme =

∗ reduces to grouped data. In this case, the (u, v)th element of the Fisher information matrix   becomes,  Iuv (θ)  

n

k

j−1 (1−ql ) l=1

j =1

qj (1−qj )

∂ qj ∂ qj , ∂θu ∂θv

for u, v = 1, 2, . . . , m. Then, Iu∗v (θ) − Iuv (θ) = n

k

j =1

j−1 (1−ql ) l=1

j−1

{1− l=1 (1−pl ) qj (1−qj )

∂ qj ∂ qj , ∂θu ∂θv

which

is a sum of rank-1 matrices, each having positive eigenvalue. Therefore Iu∗v (θ) ≥ Iuv (θ) in the sense of lower order. The statement follows. In PC-I, we begin the test with n items and k inspection times T1 < T2 < · · · < Tk . We consider the case where at each inspection time, a proportion of surviving items is removed from the test. The test is continuously monitored and the exact failure times of the items, failing in each interval are observed. This censoring scheme reduces to PIC-I if we do not record the exact times of failure, and only keep records of number of failures in each interval. We now show that non-recording of exact times of failure actually leads to loss of information. Result 2. The Fisher information of parameters of the probability distribution in a life testing experiment under PC − I scheme is more than that obtained from a similar experiment under PIC − I, where the intervals, the initial sample size and the underlying failure time distributions are identical. Proof. As defined in the introduction, when a fixed proportion of items is removed at the end of each interval, the nj items surviving at the beginning of the jth interval are i.i.d. Thus, given nj , the contributions of these items to the log-

nj 

likelihood for the jth interval is given by (see (2)),lj (θ) = l=1 δlj log qj + (1 − δlj ) log(1 − qj ) where dj and δlj as in the introduction. Let tj1 < tj2 < · · · < tjdj be the failure times of items failing in jth interval. Then given nj , the contribution of



 nj  f (tj,l ) the failures in the jth interval to the log-likelihood is lj (θ) = l=1 δlj log 1−F (Tj−1 ) + (1 − δlj ) log(1 − qj ) . Thus we have,  nj nj f (t ) lj∗ (θ) − lj (θ) = l=1 δlj log F (T )−Fj,(l T ) . Let hj (x) = F (T )−f (Fx()T ) , for j = 1, 2, . . . , k. Then lj∗ (θ) − lj (θ) = l=1 δlj log hj (tj,l ). j j j−1 j−1 ∗

This implies that

 E −

∂2 ∂θu ∂θv

1 2

3 4 5

6 7 8 9 10 11 12 13

14 15 16

17 18 19 20 21 22 23



nj

  ∗ (lj (θ) − lj (θ)) = E − l =1

∂2 δlj log hj (tj,l ) . ∂θu ∂θv 

Thus, the information matrix is a double summation over j and l, where the summand matrix happens to be the conditional information obtained from the observed failure time of the lth item when it is known to have failed in the jth interval. Since information obtained is a non-negative definite matrix, we have the stated result.

4. Simulation study Here we assess the finite sample properties of the MLEs by simulation. For the simulation study, we assume that the α lifetime follows a two parameter Weibull distribution with the distribution function F (x; α, λ) = 1 − e−(λx) ; x > 0, α > 0 and λ > 0. We choose k = 5, p1 = p2 = p3 = p4 = 0.1 and (T1 , T2 , T3 , T4 , T5 ) are pre-fixed as (0.10, 0.30, 0.50, 0.75, 0.90) quantiles of the Weibull distribution. The PIC-I data are generated (See Aggarwala Aggarwala (2001)) for n = 50, 100, 200 and 500 corresponding to (α, λ) = (1.5, 1). The average estimates (AE), mean square error (MSE), coverage percentage (CP) of asymptotic 95% confidence interval, and ratio of MSE and sample variance (S 2 ) are computed based on 10000 simulation. The results corresponding to (α, λ) = (1.5, 1) are reported in Table 1. It is evident from the table that the bias and MSE decrease as n increases. The coverage percentage of the asymptotic 95% confidence interval is satisfactory for large n. Moreover, as expected, the ratio of MSE and sample variance become closer to 1 as n increases.

24

25 26 27

28

29 30 31 32 33 34 35 36 37

8

1

S. Budhiraja et al. / Statistics and Probability Letters xx (xxxx) xxx–xxx

Acknowledgments

4

The authors would like to thank the associate editor and an anonymous referee for constructive suggestions which lead to improvement of the paper. They also like to thank Professor Anup Dewanji and Dr. Sudipta Das for their valuable suggestions and comments.

5

References

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