Reliability bounds on HNBUE life distributions with known first two moments

European Journal of Operational Research 132 (2001) 163±175

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Theory and Methodology

Reliability bounds on HNBUE life distributions with known ®rst two moments Kan Cheng a, Yeh Lam a

b,c,*

Institute of Applied Mathematics and ORSC, Academia Sinica, P.O. Box 2734, Beijing 100080, People's Republic of China b Department of Statistics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong c Lingnan (University) College, Zhongshan University, Guangzhou, People's Republic of China Received 6 November 1998; accepted 30 March 2000

Abstract In this paper, we study upper and lower bounds for the reliability function in harmonic new better than used in expectation (HNBUE) life distribution class with known ®rst two moments. Here we say a life distribution has HNBUE property if the integral harmonic mean value of the residual life in any interval ‰0; tŠ is no more than its mean. By a constructive proof, we determine the lower and upper reliability bounds analytically and show that these bounds are all sharp. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: HNBUE; Equilibrium distribution; Lower and upper bounds for the reliability function; Sharpness

1. Introduction Roughly speaking, reliability theory is concerned with determining the probability that a system will function. Thus, given a system, we may try ®rst to determine or estimate its life distribution. Alternatively, we may wish to ®nd or estimate a lower and upper bound of the system reliability. In this paper, we shall study the reliability bounds for a speci®ed class of the life distribution

* Corresponding author. Tel.: +852-26097931; fax: +85226035188. E-mail address: [email protected] (Y. Lam).

F. To state the problem more precisely, at ®rst, we de®ne the distribution function F of lifetime X as F …x† ˆ P …X < x†: Hence F …x† is a left continuous function. Then, let F …t† ˆ 1 F …t† be its survival function. Now, suppose that a life distribution class H is given, we de®ne m…t† ˆ inf F …t† and F 2H

M…t† ˆ sup F …t†: F 2H

Then m…t† is the largest lower bound of the reliability in class H, while M…t† is the smallest upper bound of the reliability in class H. Furthermore, we say that the lower bound m…t† is sharp if for each t > 0, there exists a distribution Fm …
; t† 2 H

0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 1 2 0 - X

164

K. Cheng, Y. Lam / European Journal of Operational Research 132 (2001) 163±175

such that m…t† ˆ F m …t‡; t†. Similarly, we say that the upper bound M…t† is sharp if for each t > 0, there exists a distribution FM …
; t† 2 H such that M…t† ˆ F M …t; t†. Let F be a life distribution. Denote its ®rst two R1 moments by l…F † ˆ x dF …x† and l2 …F † ˆ 0 R1 2 x dF …x†. Then the following theorem is a 0 classical result in reliability bounds due to Chebyshev (see Barlow and Marshall (1965) for reference). Theorem 1. Suppose F is a life distribution with mean l…F † ˆ 1 and l2 …F † ˆ l2 < 1. Then we have 8 0 6 t 6 1; > 2 < 1; …1 t†‡ 1 ; 1 6 t 6 l2 ; 6 F …t† 6 t 2 > l2 1 ‡ …1 t†‡ : l2 1 2 ; t P l2 : l 1‡…t 1† 2

where a‡ ˆ max…a; 0†. Furthermore, these bounds are all sharp. Barlow and Marshall (1964, 1965) studied the class in which the life distribution has an increasing failure rate (IFR) or a decreasing failure rate (DFR) with known ®rst two moments, they gave the sharp upper and lower bounds of the reliability. Later on, Marshall and Proschan (1972) considered the reliability bounds on a life distribution class with known mean, and having the property of new better than used in expectation (NBUE). Thereafter, Barlow and Proschan (1981) investigated the reliability bounds on a life distribution class with known mean and having property of increasing failure rate average (IFRA). Then Sengupta (1994) gave reliability bounds on a life distribution class with the property of new better than used (NBU). Cheng and He (1989a) studied the reliability bounds on NBUE life distributions. Recently, Cheng and Lam (1998) obtained reliability bounds on the NBUE life distributions with known ®rst two moments. Now, let X be the lifetime of a system having life distribution F. Suppose X > t P 0, and let Xt ˆ X

t

denote the residual life of a unit of age t. Thus, the mean residual life of a unit of age t is given by

E…Xt † ˆ E…X t j X > t† Z 1 F …u† du=F …t†; ˆ

t P 0:

t

…1:1†

Then, let us study the harmonic new better than used in expectation (HNBUE) class. It is a more general life distribution class. De®nition 1. Suppose that X is a non-negative random variable having distribution F with a ®nite mean l. If Z 1 1 F …u† du 6 e t=l for t P 0; …1:2† l t then X (or F ) is HNBUE. To explain the meaning of inequality (1.2), suppose for simplicity that F …x† > 0 for x > 0. Then it is straightforward to show that (1.2) is equivalent to  Z t  1 dx t 0 E…Xx †

1

6l

for t P 0:

…1:3†

This inequality says that the integral harmonic mean value of E…Xx † in 0 6 x 6 t is less than or equal to mean l. The HNBUE life distribution class was introduced by Rolski (1975). Thereafter, Klefsj o (1982) studied reliability bounds for this class; he obtained the following theorem. Theorem 2. Suppose that F 2 HNBUE and l…F † ˆ 1. Then we have m…t† 6 F …t† 6 M…t† ˆ e where m…t† ˆ





e a; 0;

…t 1†‡

;

…1:4†

0 6 t 6 1; t P 1;

and a ˆ a…t† P t is the unique solution of ut …a† ˆ 0 with ut …a† ˆ …a ‡ 1

t†e

a

1 ‡ t:

…1:5†

Furthermore, the lower and upper bounds are all sharp.

K. Cheng, Y. Lam / European Journal of Operational Research 132 (2001) 163±175

In this paper, we study the reliability bounds on the HNBUE class with known ®rst two moments. The upper and lower bounds of the reliability function are determined analytically for ®ve subregions. For the proof of sharpness, in each subregion, we construct a distribution in the HNBUE class with known ®rst two moments, and show that it will attain the lower or upper bound. In Section 2, we shall study a speci®c second moment q…F † and its properties. In Section 3, the lower and upper reliability bounds on the HNBUE life distributions with known ®rst two moments are studied through three propositions. The results are then summarized as Theorem 3 which is a generalization and improvement of Theorems 1 and 2. As a simple application, Theorem 3 is applied to the approximation problem for the HNBUE class in Section 4.

165

This implies that q…F † P 0. 3. q…F † ˆ 1=2, if and only if F is degenerate at l…F †. 4. Assume that F 2 HNBUE, then q…F † ˆ 0, if and only if F is an exponential distribution. From (2.1), Property 3 is trivial. For the proof of Property 4, see Basu and Kirmani (1986). A more general condition for F 2 HNBUE to be an exponential distribution was given by Bhattacharjee and Sethuraman (1990). By using l…F † and q…F † as a `replacement' of the ®rst two moments l…F † and l2 …F †, we are interested in the following two problems. 1. Hl …q† problem: given that t > 0 and 0 < q < 1=2, and Hl …q† ˆ fF : F 2 HNBUE; l…F † ˆ l; q…F † ˆ qg; ®nd

2. q…F† and its properties

ml …t; q† ˆ

To study the reliability bounds for any life distribution F in the HNBUE class with known ®rst two moments, l…F † ˆ l and l2 …F † ˆ l2 say. It is more convenient to use mean l…F † and q…F † ˆ 1

l2 …F † 2l…F †

2

:

…2:1†

This is because l…F † and q…F † are equivalent to l…F † and l2 …F †. This is also because q…F † has some nice properties. 1. q…F † is invariant under scale transformation. Assume that X  F and cX  G, i.e., X is distributed with distribution F and cX is distributed with distribution G, where c 6ˆ 0 is a constant, then q…F † ˆ q…G†. 2. If F 2 HNBUE, then 0 6 q…F † 6 1=2. To see this, we note that the inequality 2 l2 …F † P l…F † implies that q…F † 6 1=2. Moreover, by using (1.2), we have Z 1 Z 1 2 l2 …F † ˆ x dF …x† ˆ 2 xF …x† dx 0 0 Z 1Z 1 Z 1 ˆ2 F …x† dx dt 6 2l e t=l dt 0

t

ˆ 2l2 ˆ 2l…F †2 :

0

…2:2†

inf

F 2Hl …q†

F …t† and

Ml …t; q† ˆ sup F …t†; F 2Hl …q†

then justify the sharpness. 2. H…q† problem: given that t > 0 and 0 < q < 1=2, and H…q† ˆ fF : F 2 HNBUE; l…F † ˆ 1; q…F † ˆ qg;

…2:3†

®nd m…t; q† ˆ inf F …t† F 2H…q†

and

M…t; q† ˆ sup F …t†; F 2H…q†

…2:4†

then justify the sharpness. Although the H…q† problem is a special case of the Hl …q† problem, both problems are equivalent. This is due to the following relationships:   t ;q and ml …t; q† ˆ m l   t Ml …t; q† ˆ M ;q : …2:5† l Thus, we can concentrate our attention on the study of the H…q† problem. Note that if l ˆ 1, then

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K. Cheng, Y. Lam / European Journal of Operational Research 132 (2001) 163±175

q…F † ˆ 1 ˆ1

1 l …F † ˆ 1 2 2 Z 1 G…x† dx; 0

R1

Z

1 0

3. The reliability bounds

xF …x† dx; …2:6† R1

where G…x† ˆ x F …u† du=l ˆ x F …u† du is the equilibrium distribution of F, and G…x† ˆ 1 G…x† (see Ross (1996) for reference). To solve the H…q† problem, according to Properties 2±4 of q…F †, we only need to study the reliability bounds m…t; q† and M…t; q† in the following region:   1 W ˆ …t; q† : t > 0; 0 < q < : …2:7† 2 Lemma 1 is a simple but useful result. Lemma 1. Let F and G be two life distributions satisfying 1. l…F † ˆ l…G†; l2 …F † and l2 …G† are both ®nite; 2. there exists t > 0 such that F …x† P G…x†;

0 6 x 6 t;

…2:8†

F …x† 6 G…x†;

x > t:

…2:9†

Then l2 …F † 6 l2 …G† and

q…F † P q…G†:

…2:10†

In addition, if there exists a set with a positive measure such that one of inequalities (2.8) and (2.9) holds in the strict sense, then inequalities (2.10) will also hold in the strict sense. The proof is straightforward. In fact, from (2.2) we have Z 1  Z 1 l2 …G† l2 …F † ˆ 2 xG…x† dx xF …x† dx 0 0 Z 1  ˆ2 …x t†‰G…x† F …x†Š dx Z ˆ2 Z ‡

0

t 0 1

t

Hence, (2.10) follows.

…x

t†‰G…x† t†‰G…x†

1 IAA …t† ˆ ‰1 2 1 IAC …t† ˆ ‰1 2 1 IBB …t† ˆ ‰1 2

t†a Š;

…1 a

e …t

0 < t < 1;

…1

t†a Š;

2

1

1† =…et

…3:1†

0 < t < 1; …3:2†

1†Š;

t P 1;

…3:3†

and 1 IBC …t† ˆ e 2

…t 1†

;

t P 1;

…3:4†

where a ˆ a…t† P t is the unique solution of ut …a† ˆ 0 given in Theorem 2. It is easy to show that for 0 < t < 1, 1 0 < IAC …t† < IAA …t† < ; 2 while for t P 1, 1 0 < IBC …t† < IBB …t† < : 2 Then we shall partition region W into ®ve subregions (see Fig. 1), namely W ˆ A1 [ A2 [ B1 [ B2 [ C; where A1 ˆ f…t; q† : 0 < t < 1; IAC …t† 6 q < IAA …t†g;   1 ; A2 ˆ …t; q† : 0 < t < 1; IAA …t† 6 q < 2   1 ; B1 ˆ …t; q† : t > 1; IBB …t† 6 q < 2 B2 ˆ f…t; q† : t > 1; IBC …t† 6 q < IBB …t†g; C ˆ f…t; q† : 0 < t < 1; 0 < q < IAC …t†; or t P 1; 0 < q < IBC …t†g:

F …x†Š dx 

…x

First of all, de®ne

F …x†Š dx P 0:

Now we have the following results. Proposition 1. For F 2 H…q† and …t; q† 2 A ˆ A1 [ A2 , we have

K. Cheng, Y. Lam / European Journal of Operational Research 132 (2001) 163±175

167

Fig. 1. Partitions of region W for reliability bounds.

e

a

…t; q† 2 A1 ; …t; q† 2 A2 ;

…1 t†2 …1 t†2 ‡1 2q

) 6 F …t† 6 1;

…3:5†

where a is given in Theorem 2. Moreover, these lower and upper bounds are sharp. Proof. (1) The lower bound. First of all, assume that …t; q† 2 A1 . By Theorem 2, we only need to justify the sharpness. To do this, de®ne a life distribution F1 whose survival function is given by 8 0 6 x 6 t; < 1;  F 1 …x; t; a† ˆ e a ; t < x 6 a; …3:6† : be x ; x > a; where a and b are constants determined by conditions l…F1 † ˆ 1 and a 6 a 6 a ‡ 1. The condition l…F1 † ˆ 1 yields b ˆ …a ‡ 1

a†ea

a

:

…3:7†

It is clear that 0 6 b 6 1. Furthermore, let the equilibrium distribution of F1 be G1 . Then Z 1 G1 …x† ˆ F 1 …u; t; a† du x 8 0 6 x 6 t; < 1 x;  ˆ be a ‡ …a x†e a ; t < x 6 a; : be x ; x > a: Direct veri®cation shows that G1 …x† 6 e x ; x P 0. For example, for t < x 6 a, we have by using (3.7) G1 …x† ˆ be ˆe

a a

‡ …a

…1 ‡ a

x†e

a

x† 6 e

a


e

a

x

ˆ e x:

The proof for the other intervals is similar. Thus, F1 2 HNBUE. Now, (2.6) with the help of (3.7) yields Z q…F1 † ˆ 1

1 0

1 ˆ f1 e 2 ˆ At …a†:

G1 …x† dx a

a …1

t† ‡ …a

2

a † e

a

g

Obviously, for 0 < t < 1; At …a† is continuous and strictly increasing in a 2 ‰a ; a ‡ 1Š. Moreover, it is easy to see that At …a † ˆ IAC …t† and At …a ‡ 1† ˆ IAA …t†. Therefore, for …t; q† 2 A1 , At …a† ˆ q has a unique solution a . For this solution a , the corresponding distribution F1 …x; t; a † will satisfy q…F1 …
; t; a †† ˆ q; hence F1 …
; t; a † 2 H…q†. Furthermore F 1 …x; t; a †jxˆt‡ ˆ e

a

:

Thus, the lower bound is sharp in A1 . Now, assume that …t; q† 2 A2 . From Theorem 1, we again only need to prove the sharpness. For this purpose, let us denote F …t† ˆ c 6 1 for a certain value of t, and ®rst assume that cPe

a

:

…3:8†

Later we shall show that this inequality always holds. Then a life distribution F2 is constructed whose survival function is given by

168

K. Cheng, Y. Lam / European Journal of Operational Research 132 (2001) 163±175

8 < 1; F 2 …x; t; c† ˆ c; : 0;

0 6 x 6 t; t < x 6 t ‡ 1c …1 t†; x > t ‡ 1c …1 t†:

…3:9†

It is easy to verify that l…F2 † ˆ 1. Furthermore, let the equilibrium distribution of F2 be G2 . Then Z 1 G2 …x† ˆ F 2 …u; t; c† du x

8 < 1 x; ˆ c‰t ‡ 1c …1 : 0;

t†

0 6 x 6 t; t 6 x 6 t ‡ 1c …1 t†; x P t ‡ 1c …1 t†:

xŠ; x

Then we show that G2 …x† 6 e , x P 0. In fact, this result will hold if we can prove that for 0 < t < 1,   1 h…x† ˆ e x c t ‡ …1 t† x P 0; c 1 t 6 x 6 t ‡ …1 t†: …3:10† c It is clear that h…x† is convex and will take the minimum at x ˆ `n c. Therefore, inequality (3.10) will hold if h…x † P 0 for 0 < t < 1. Note that h…x † ˆ …c e

a

1†…1

t†

c
`n c ˆ w…c†;

6 c 6 1:

Since w…c† is concave and hence it takes the min imum at either e a or 1. However, it follows from (1.5) that w…e

a

† ˆ …e

a

1†…1

t† ‡ a e

a

ˆ ut …a † ˆ 0

and w…1† ˆ 0. This implies that the minimum of w…c† is 0. Therefore, h…x † ˆ w…c† P 0;

0 < t < 1:

This completes the proof of G2 …x† 6 e x for x P 0. Consequently, F2 2 HNBUE. Moreover, we have from (1.5) that Z 1 q…F2 † ˆ 1 G2 …x† dx 0   1 1 1 2 1 ˆ ‡ …3:11† …1 t† ˆ Bt …c†: 2 2 c

Obviously, Bt …c† is strictly increasing in c. Note that it follows from (1.5) that Bt …e

a

1 † ˆ ‰1 …1 2 ˆ Bt …1†:

t†a Š ˆ IAA …t† 6 q <

1 2

Consequently, there exists a unique solution  c 2 ‰e a ; 1†, such that Bt …c † ˆ q. Then it is trivial from (3.11) and Bt …c † ˆ q that c ˆ

…1 …1

t†2

2

t† ‡ 1

2q

:

Theorem 1 implies that c is a lower bound of the reliability in A2 . Now since F2 …
; t; c † 2 H…q† and F 2 …x; t; c †jxˆt‡ ˆ c : Therefore, the lower bound is also sharp in A2 , so  is in A. Note that in virtue of the fact c 2 ‰e a ; 1†, the assumption (3.8) does not lose generality. (2) The upper bound. Suppose that …t; q† 2 A ˆ A1 [ A2 , as 1 is clearly an upper bound of the reliability on the subregion A, again, we only need to prove the sharpness. It follows from (1.5) and (3.2) that IAC …t† is continuous and strictly increasing in t. Furthermore, for …t; q† 2 A, we can see from (1.5) that 1 IAC …t† 6 q < IAC …1† ˆ : 2 Thus, there exists a unique t0 such that IAC …t0 † ˆ q

and

0 < t 6 t0 < 1:

Now, consider a life distribution F3 whose survival function is given by 8 < 1;  F 3 …x; t0 † ˆ e a0 ; : x e ;

0 6 x 6 t0 ; t0 < x 6 a0 ; x P a0 ;

…3:12†

where a0 ˆ a…t0 † is the unique solution of ut0 …a† ˆ 0 de®ned in Theorem 2. It follows from ut0 …a0 † ˆ 0 that l…F3 † ˆ 1. Let the equilibrium distribution of F3 be G3 . Then

K. Cheng, Y. Lam / European Journal of Operational Research 132 (2001) 163±175

Z G3 …x† ˆ

1 x

F 3 …u; t0 † du

8 < 1 x;  ˆ e a0 …1 ‡ a0 : x e ;

x†;

0 6 x 6 t0 ; t0 < x 6 a0 ; x P a0 :

It is easy to show that G3 …x† 6 e x , for x P 0. This implies that F3 2 HNBUE. Furthermore, we have Z 1 G3 …x† dx q…F3 † ˆ 1 0

1 ˆ f1 2

e

a0

…1

t0 †a0 g ˆ IAC …t0 † ˆ q:

Therefore F3 …
; t0 † 2 H…q†. Now, because t 6 t0 , it follows from (3.12) that F 3 …x; t0 †jxˆt ˆ 1: Thus, the upper bound is sharp in A. This completes the proof of Proposition 1. Proposition 2. For F 2 H…q† and …t; q† 2 B ˆ B1 [ B2 , we have ( 1 2q ; …t; q† 2 B1 ; 2 2q 0 6 F …t† 6 …t …t1† ‡1 …3:13† 1† e ; …t; q† 2 B2 : In addition, the lower and upper bounds are all sharp. Proof. (1) The lower bound. Since 0 is a lower bound, we only need to show that the lower bound 0 is sharp. To do this, a life distribution F4 is constructed whose survival function is given by 8 x < e ; 0 6 x 6 `n…2q†; `n…2q† < x 6 `n…2q† ‡ 1; F 4 …x† ˆ 2q; : 0; x > `n…2q† ‡ 1; …3:14† where `n…2q† > 0. Obviously, we have l…F4 † ˆ 1. By direct veri®cation, it is easy to show that F4 2 HNBUE, and q…F4 † ˆ q. On the other hand, since …t; q† 2 B, then q P …1=2†e …t 1† , or t P `n…2q† ‡ 1. As a result, F4 2 H…q†

and

F 4 …x†jxˆt‡ ˆ 0:

Hence, the lower bound 0 is sharp in B.

169

(2) The upper bound. At ®rst, assume that …t; q† 2 B1 . Then for any F 2 H…q†, denote F …t† ˆ c. It follows from Theorem 2 that 0 6 c 6 e …t 1† . Now, a life distribution F5 is constructed such that its survival function is given by 8 < 1; 0 6 x 6 a; F 5 …x; a† ˆ c; a < x 6 t; …3:15† : 0; x > t; where a ˆ …1 ct†=…1 c†. It is easy to justify that l…F5 † ˆ 1 and 0 6 a 6 1. Moreover, by a direct veri®cation, we can show that F5 2 HNBUE, and i 1h c 2 …t 1† : q…F5 † ˆ 1 2 1 c On the other hand, it is clear from (3.15) that F 5 …x; a† P F …x† for 0 6 x 6 a and F 5 …x; a† 6 F …x† for x > a. Then Lemma 1 implies that q…F5 † P q…F † ˆ q, i.e., i 1h c 1 …t 1†2 P q: …3:16† 2 1 c The solution to inequality (3.16) is given by c 6 c ˆ

1 …t

2q 2

1† ‡ 1

2q

:

Consequently, c is an upper bound in B1 . The equality q…F5 † ˆ q holds, if and only if c ˆ c . Thus, let a ˆ

1 1

c t : c

Then the corresponding distribution function F5 …x; a † will satisfy q…F5 …
; a †† ˆ q. Therefore, F 5 …
; a † 2 H…q†

and

F 5 …x; a †jxˆt ˆ c :

Thus, the upper bound c is sharp in B1 . Now, assume that …t; q† 2 B2 . According to Theorem 2, again we only need to prove the sharpness. To do this, at ®rst denote c0 ˆ e …t 1† < 1, and for t > 1, let a0 ˆ

1 1

c 0 t et ˆ t c0 e

1 1

Clearly, 0 < a0 < 1.

t : 1

…3:17†

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K. Cheng, Y. Lam / European Journal of Operational Research 132 (2001) 163±175

Now, for ®xed a 2 ‰0; a0 †, de®ne a function F6 such that F 6 …x; a† ˆ 1 F6 …x; a† 8 1; 0 6 x 6 a; > > < x e ; a < x 6 b; ˆ c; b < x 6 t; > > : 0 0; x > t;

…3:18†

b

e

‡ a ‡ c0 …t

b† ˆ 1:

Now, let b ˆ bt …a† be the solution to the equation Aa …b† ˆ 0, where Aa …b† ˆ 1

a

‰e

Because c0 ˆ e Aa …a† ˆ …c0 > …c0 Aa …t

1† ˆ

b

e

…t 1†

‡ a ‡ c0 …t

b†Š:

a

…e

…3:19†

< 1, from (3.19) we have

1†a ‡ 1 c0 t 1†a0 ‡ 1 c0 t ˆ 0; …1

…3:20†

a†† 6 0:

…3:21†

Note that (3.17) implies that a0 < t 1. Then, the combination of (3.20) and (3.21) gives a < b ˆ bt …a† 6 t

1:

…3:22†

Therefore, (3.18) and (3.22) yield that F6 is actually a life distribution with mean l…F6 † ˆ 1. Denote its equilibrium distribution by G6 . Then Z 1 G6 …x† ˆ F 6 …u; a† du x 8 1 x; 0 6 x 6 a; > > < x e e b ‡ c0 …t b†; a 6 x 6 b; ˆ c …t x†; b 6 x 6 t; > > : 0 0; x P t: It is trivial to show that G6 …x† 6 e x ; x P 0. As an example, for a 6 x 6 b, we have G6 …x† ˆ e

x

6e

x

e

b

e

b

‡ c0 …t ‡e

b†

…t 1† t b 1

e

x

1 0

G6 …x† dx

1 ˆ a2 ‡ …b a†…e a 1 ‡ a† 2 1 2 ‡ c0 ‰1 …1 t ‡ b† Š ˆ Ct …a†: 2

db ˆ …1 da

0

a

q…F6 † ˆ 1

Obviously, Ct …a† is continuous in a 2 ‰0; a0 †. By di€erentiating Aa …b† ˆ 0 with respect to a, we obtain

where b is determined by the condition Z 1 l…F6 † ˆ F 6 …x; a† dx ˆe

Z

x

ˆe :

The proof of G6 …x† 6 e in other intervals is similar. Therefore, F6 2 HNBUE. Furthermore,

e a †=…c0

e b † < 0;

…3:23†

since b < t 1. Then by using (3.19) and (3.22), it follows that Ct0 …a† ˆ …1

e a †…b

a† > 0:

Consequently, Ct …a† is strictly increasing in a 2 ‰0; a0 †. If a ˆ 0, equation A0 …b† ˆ 0 gives b ˆ t 1, and 1 Ct …0† ˆ e 2

…t 1†

ˆ IBC …t†;

t > 1:

On the other hand, it follows from (3.19) with the help of (3.17) that if a ! a0 , then Aa0 …a0 † ! 0, and we have b ! a0 . As a result, it follows that 1 1 Ct …a0 † ˆ a20 ‡ c0 ‰1 …1 2 2 1 ˆ ‰1 …t 1†2 =…et 2 t > 1:

2

t ‡ a0 † Š 1

1†Š ˆ IBB …t†;

In other words, for t > 1, IBC …t† ˆ Ct …0† 6 Ct …a† < Ct …a0 † ˆ IBB …t†: Thus, for any …t; q† 2 B2 , there exists a unique a 2 ‰0; a0 † satisfying Ct …a † ˆ q. Then, for such a ; F6 …
; a † 2 H…q† and F 6 …x; a †jxˆt ˆ c0 ˆ e

…t 1†

:

This means that the upper bound e …t 1† is sharp in B2 . This completes the proof of Proposition 2.

K. Cheng, Y. Lam / European Journal of Operational Research 132 (2001) 163±175

Proposition 3. Assume that F 2 H…q† and …t; q† 2 C. Then, we have …b ‡c †

e

6 F …t† 6 e

b

t†…ec

1† ˆ c

b

 1

e

c

ec

1

ˆ q:

1 ˆ e 2

b

1

e

c



c2 ec

1

:

c < e t:

Proof. First of all, for …t; q† 2 C, we construct a special distribution F7 2 H…q†. To do this, de®ne a function F7 such that

where b and c satisfy 0 < b < t, t Z 1 F 7 …u; t; b; c† du ˆ 1:

G7 …x† dx

…3:25†

Furthermore, the lower and upper bounds are both sharp.

F 7 …x; t; b; c† ˆ 1 F7 …x; t; b; c† 8 x e ; 0 6 x 6 b; > > < b e ; b 6 x 6 t; ˆ e …b‡c† ; t < x 6 b ‡ c; > > : x e ; x P b ‡ c;

0



…3:24† 

c2

1

Now, for a life distribution F 2 H…q† and …t; q† 2 C, we shall study its lower and upper bounds, respectively. (1) The lower bound. For F 2 H…q† and …t; q† 2 C, let F …t† ˆ c and assume tentatively that

and 1 e 2

q…F7 † ˆ 1

;

where b and c are the unique positive solution of the following equations: …1 ‡ b

Z

171

…3:26†

b < 1, and

0

The last equation implies that Eq. (3.24) holds. Therefore, it follows from (3.24) that 0 < c= …ec 1† < 1, hence c must be positive. Moreover, b ‡ c ˆ t ‡ …1 ec …1 c††=…ec 1† > t. This implies that function F7 is actually a distribution function. Let the equilibrium distribution of F7 be G7 . Then we have Z 1 G7 …x† ˆ F 7 …u; t; b; c† du x 8 x e ; 0 6 x 6 b; > > < …1 ‡ b x†e b ; b 6 x 6 t; ˆ …1 ‡ b ‡ c x†e …b‡c† ; t 6 x 6 b ‡ c; > > : x e ; x P b ‡ c: Moreover, it is easy to verify that G7 …x† 6 e x , for x P 0. Consequently, F7 2 HNBUE with

…3:27†

Later we shall show that inequality (3.27) always holds. Thus, by noting that b ‡ c > t, we can choose b and c satisfying (3.24) and cˆe

…b‡c†

:

…3:28†

Therefore, b and c are functions of c, b ˆ b1 …c† and c ˆ c1 …c† say. Furthermore, q…F7 † is also a function of c, and we can write   1 b c2 c wt …c† ˆ q…F7 † ˆ e 1 e : …3:29† 2 ec 1 It follows from Theorem 2 that F …t† ˆ c P m…t†. Thus, in combination with (3.27) we have m…t† 6 c < e t :

…3:30†

If c ! e t , then (3.24) and (3.28) imply that b ! t; c ! 0 and wt …e t † ˆ 0.  If 0 < t < 1, letting c ! m…t† ˆ e a , (1.5) together with (3.24) and (3.28) shows that b ! 0; c ! a . Thus, (3.29) yields that ! 2 1 a a a 1 e wt …e † ˆ 2 ea 1 1 ˆ ‰1 2

e

a

…1

t†a Š ˆ IAC …t†:

However, if t P 1, letting c ! m…t† ˆ 0, then (3.24) and (3.28) imply that b ! t 1; c ! ‡1. Consequently, we have 1 wt …0† ˆ e 2

…t 1†

ˆ IBC …t†:

On the other hand, it follows from (3.24) and (3.28) that

172

K. Cheng, Y. Lam / European Journal of Operational Research 132 (2001) 163±175

db e c …1 ˆ dc c…ec 1

c† > 0; c†

2

db e b ‰ec …c 1† ‡ 1Š dc < 0: wt …c† ‡ 2 dc dc 2ec …ec 1†

ˆ

Thus, wt …c† is strictly decreasing and continuous in ‰m…t†; e t †. In conclusion, for c 2 ‰m…t†; e t †, we have  I …t†; 0 < t < 1; 0 < wt …c† 6 AC …3:31† IBC …t†; t P 1: Thus, for any …t; q† 2 C, there exists a unique solution cm satisfying wt …cm † ˆ q and m…t† 6 cm < e t :

…3:32†

Clearly, cm is a function of t. Now, let the equilibrium distribution of F be denoted by G. Since F 2 HNBUE and l…F † ˆ 1, we have G…x† 6 e x ;

x P 0:

…3:33†

In the following, we shall prove that G…x† 6 G7 …x†:

…3:34†

In fact, (3.34) holds for 0 6 x 6 b and x P b ‡ c. For t 6 x 6 b ‡ c, we have F …x† 6 F …t† ˆ c, then (3.28) and (3.32) imply that Z 1 G…x† ˆ F …u† du x

Z ˆ

x

‡

b‡c

Z

F …u† du

1

b‡c

6 c…b ‡ c

wt …c† ˆ q…F7 † Z ˆ1

…b‡c†

x† ‡ G…b ‡ c†

ˆ G7 …x†:

Now, for b 6 x 6 t, there exists 0 6 k 6 1 such that x ˆ kb ‡ …1 k†t. Then, by noting that G…x† is convex, and G7 …x† is linear in ‰b; tŠ, we have

1

0

Z G7 …x† dx 6 1

1 0

ˆ q…F † ˆ q ˆ wt …cm †:

G…x† dx …3:35†

Because wt …c† is decreasing in c, (3.34) implies that c P cm . This means that cm is a lower bound of the reliability in subregion C. Furthermore, as cm is the unique solution to equation wt …c† ˆ q, we conclude that b ˆ b1 …cm † and c ˆ c1 …cm † are the unique positive solution to (3.24) and (3.25). Since cm is a function of t, b and c are also functions of t, b ˆ b…t† and c ˆ c…t† say. In conclusion, we have F 7 …
; t; b ; c † 2 H…q†; and by (3.28), cm ˆ e F 7 …x; t; b ; c †jxˆt‡ ˆ e

…b ‡c † 

…b

, then

‡c †

:





Thus, the lower bound e …b ‡c † is sharp in subregion C. Note that from (3.32) we always have cm < e t , hence the assumption F …t† ˆ c < e t does not lose generality. This completes the proof for the lower bound case. (2) The upper bound. Now, for F 2 H…q† and …t; q† 2 C, let F …t† ˆ c, and assume tentatively that F …t† ˆ c > e t :

…3:36†

Later we shall show that (3.36) always holds. By applying Theorem 2, we then have e

F …u† du 6 c…b ‡ c x† ‡ e

k†G7 …t† ˆ G7 …x†:

This completes the proof of (3.34). Therefore,

Therefore, (3.29) implies that w0t …c†

k†G…t†

6 kG7 …b† ‡ …1

1†2 < 0: 1 c†

…ec c ce …ec

dc ˆ dc

G…x† 6 kG…b† ‡ …1

t

< c6e

…t 1†‡

:

…3:37†

Then consider the life distribution F7 de®ned by (3.26). As 0 < b < t, we can choose b satisfying c ˆ e b:

…3:38†

Therefore, b and c are functions of c, b ˆ b2 …c† and c ˆ c2 …c† say. Under condition (3.38), let us study function wt …c† again.

K. Cheng, Y. Lam / European Journal of Operational Research 132 (2001) 163±175

If 0 < t < 1, letting c ! e …t 1†‡ ˆ 1, then (3.38) shows that b ! 0, and (3.24) with the help of (1.5) yields that c ! a . Hence, wt …e

…t 1†‡

1 † ˆ ‰1 2

a

e

…1 …t 1†‡

t†a Š ˆ IAC …t†:

If t P 1, letting c ! e ˆe from (3.24) and (3.38) that b ! t As a result, we have

…t 1†

, we can see 1 and c ! 1.

173

Then (3.39) can be rewritten as wt …c† 6 wt …cM † for c 2 …e t ; e

…t 1†‡

Š:

Since wt …c† is strictly increasing in c, it follows that F …t† ˆ c 6 cM :

By a similar argument as applying to the proof of the lower bound case, we can show that

In other words, cM is an upper bound of F …t† in C. Due to the fact that cM is the unique solution of (3.41) in c 2 …e t ; e …t 1†‡ Š, then b2 …cM † and c2 …cM † must be the solution of (3.24) and (3.25), and they are positive. It has been shown in the proof of the lower bound case that b and c are the unique positive solution of these two equations. Therefore, we have b ˆ b2 …cM † and c ˆ c2 …cM †. As a  result, (3.38) shows that cM ˆ e b is an upper bound in subregion C. Now, F7 …
; t; b ; c † 2 H…q†, and from (3.26) we have

wt …c† ˆ q…F7 † 6 q…F † ˆ q:

F 7 …x; t; b ; c †jxˆt ˆ e

wt …e

…t 1†‡

1 †ˆ e 2

…t 1†

ˆ IBC …t†:

However, if c ! e t ‡, we shall have b ! t; c ! 0, and then wt …e t ‡† ˆ 0:

…3:39†

1†2 > 0; 1† ‡ 1Š

Furthermore,  1 0 wt …c† ˆ 1 e 2 ‡

c

c2 ec

c‰e …c 2ec …ec



By combining Propositions 1±3, we have obtained the following theorem.

1

Theorem 3. have

1† ‡ 1Š dc > 0: 2 dc 1†

Consequently, from (3.39) and (3.40), there exists a unique cM such that

t

< cM 6 e

where

m…t; q† ˆ

and

…3:41†

and e

M…t; q† ˆ …t 1†‡

:

For F 2 H…q†, and …t; q† 2 W , we

m…t; q† 6 F …t† 6 M…t; q†;

Thus, wt …c† is continuous and strictly increasing in c 2 …e t ; e …t 1†‡ Š. To sum up, for …t; q† 2 C, we have  I …t†; 0 < t < 1; 0 < wt …c† 6 AC …3:40† IBC …t†; t P 1:

wt …cM † ˆ q



c > 0:

2

c

;

i.e., the upper bound e b is sharp in C. Due to inequality (3.42), assumption (3.36) does not lose of generality. This completes the proof for the upper bound case, so does the proof of Proposition 3.

Now, it is straightforward from (3.24) and (3.38) that dc …ec ˆ c dc c‰e …c

b

…3:42†

8 e > > > <

a

;

…1 t†2 ; …1 t†2 ‡1 2q

…t; q† 2 A1 ; …t; q† 2 A2 ;

> > 0; > : …b ‡c † e ;

…t; q† 2 B ˆ B1 [ B2 ; …t; q† 2 C;

8 1; > > > <

…t; q† 2 A ˆ A1 [ A2 ; …t; q† 2 B1 ;

1 2q ; …t 1†2 ‡1 2q > …t 1†

e > > : e

b

;

;

…t; q† 2 B2 ; …t; q† 2 C:

174

K. Cheng, Y. Lam / European Journal of Operational Research 132 (2001) 163±175

Here, a ˆ a…t† P t is the unique solution of equation …a ‡ 1

t†e

a

1 ‡ t ˆ 0;

…3:43†

while b ˆ b…t† and c ˆ c…t† are the unique positive solution of the equations …1 ‡ b

t†…ec

1† ˆ c;

…3:44†

and 1 e 2

b

 1

e

c



c2 ec

1

ˆ q:

…3:45†

Furthermore, the lower and upper bounds m…t; q† and M…t; q† are all sharp. Theorem 3 is the main result of this paper. It is a generalization of Theorems 1 and 2. This is because Theorem 1 studies the reliability bounds for the class with known ®rst two moments only, while Theorem 2 considers the reliability bounds for HNBUE class with known ®rst moment. However, our results are concerned with the reliability bounds for HNBUE class with known ®rst two moments, which clearly is a subclass of the former two classes studied in Theorems 1 and 2, respectively.

4. A simple application As a simple application of our result, we consider the approximation problem of a life distribution by the exponential distribution. Namely, we wish to study the upper bound of the approximation K…H† ˆ

sup jF …t†

F 2H;t P 0

e t j:

As an alternative approach, Theorem 3 can be applied to solve the approximation problem for the life distribution class H…q†. Theorem 4. For F 2 H…q†, we have K…q† ˆ

sup

F 2H…q†; t P 0

e tj ˆ 1

jF …t†

e

t1

;

where t1 is determined by IAC …t1 † ˆ q. Furthermore, the upper bound of the approximation K…q† is sharp. Proof. Since m…t; q† ˆ inf F 2H…q† F …t†; M…t; q† ˆ supF 2H…q† F …t†, it is trivial that e t j; jm…t; q†

K…q† ˆ sup …jM…t; q† tP0

e t j†:

Then it is easy to check that K…q† ˆ max…1

e

t1

;e

b1

e

t1

;e

b2

e

t2

†;

where t1 is the solution of equation IAC …t1 † ˆ q; t2 is the solution of equation IBC …t2 † ˆ q, and b1 ˆ b…t1 †; b2 ˆ b…t2 † are the solutions of (3.44) and (3.45) for t ˆ t1 and t2 , respectively. It is easy  to check that e b …t† e t is decreasing in t 2 ‰t1 ; t2 Š. Thus we have K…q† ˆ 1

e

t1

:

To prove the sharpness, we consider distribution function F3 …t; t1 † in Proposition 1. Taking F …t† ˆ F3 …t; t1 † and t ˆ t1 , then jF 3 …t; t1 † e t j tˆt ˆ 1 e t1 ˆ K…q†: 1

Although Theorem 4 has been obtained by Daley (1988), it is a straightforward conclusion of our results here.

…4:1†

An interesting problem is to determine the value K…H† for some speci®c life distribution class. Daley (1988) considered the problem for NBUE and HNBUE classes and their duals. Cheng and He (1988, 1989b, 1991) discussed the same problem for NBUE, IFR and some other life distribution classes.

Acknowledgements The authors are grateful to the editor and referees for their valuable comments and suggestions. Cheng's research was supported by a direct grant from the Chinese University of Hong Kong (project code: 2060155), and partially supported by the National Natural Science Foundation of China.

K. Cheng, Y. Lam / European Journal of Operational Research 132 (2001) 163±175

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