THE NBUT CLASS OF LIFE DISTRIBUTIONS
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THE NBUT CLASS OF LIFE DISTRIBUTIONS I. A. Ahmad∗, M. Kayid †and X. Li‡ IEEE Transaction on Reliability, to appear, June, 2005
Abstract A new class of life distributions, namely new better than used in the total time on test transform ordering (N BU T ) is introduced. The relations of this class to other classes of life distributions, closure properties under some reliability operations are discussed as well. We provide a simple argument based on stochastic orders that the family of the N BU T distributions is closed under the formation of series systems in case of independent identically distributed components. Behavior of this class is developed in terms of the monotonicity of the residual life of k−out of−n systems given the time of the (n − k)-th failure. Finally, testing exponentially against N BU T class is discussed
Keywords: T T T transform order, stochastic order, increasing concave order, random minima, series system, mixing, k out-of-n systems, life testing
1
Introduction and motivation
Since the accurate distribution of the life of an element or a system is often unavailable in practical situation, nonparametric ageing properties have been found to be quite useful in modeling ageing or wear-out process and to conduct maintenance policy in reliability. Such ageing classes are derived via several notions of comparison between random variables. Of the most commonly used comparison we find, cf. Muller and Sotayan (2002) and Shaked and Shanthikumar (1994), the stochastic comparison and the increasing concave comparison. Formally, if X and Y are two random variables with distributions F and G (survivals F and G), respectively, then we say that X is smaller than Y in the: ∗
Dept. of Stat. and Act. Science, University of Central Fl., Orlando, Florida 32816-2370, USA Dept. of Mathematics, Faculty of Education (Suez ), Suez Canal University, EGYPT ‡ Dept. of Mathematics, Lanzhou University, Lanzhou 730000, People’s Republic of CHINA †
1
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(i) stochastic order (denoted by X ≤st Y ), if E [φ(X)] ≤ E [φ(Y )]
for all increasing functions φ;
(ii) increasing concave order (denoted by X ≤icv Y ), if E[φ(X)] ≤ E[φ(Y )] for all increasing concave functions φ. Consider a distribution function F , of a non-negative random variable X, which is strictly increasing on its interval support. Let p ∈ (0, 1) and t ≥ 0 be two values related by t = F −1 (p), where F −1 is the right continuous inverse of F. Denote by ª © AF ≡ (x, u) : u ∈ (0, p), x ∈ (0, F −1 (u)) , and
ª © BF ≡ (x, u) : u ∈ (p, 1), x ∈ (0, F −1 (u)) .
The above areas of the regions have various intuitive meanings in different applications. For example, if F is the distribution function of the lifetime of a machine, then TX (p) ≡ kAF (p) ∪ BF (p)k , p ∈ (0, 1), corresponds to the total time on test (T T T ) transform associated with this distribution. Recently the total time on test (T T T ) transform order has also come to use in reliability and life testing (see, Kochar et al. (2002)). We say that a random variable X is smaller than a random variable Y in the T T T -transform order (denoted by X ≤ttt Y ) if, and only if, TX (p) =
Z
0
F −1 (p)
F (x) dx ≤
Z
G−1 (p)
G(x) dx = TY (p),
0
p ∈ (0, 1),
where F −1 and G−1 are respectively the right continuous inverses of F and G. The T T T -transform order is weaker than the stochastic order (≤st ) but is stronger than the increasing concave order (≤icv ). For the previous orders we have the following relationships (see, Kochar et al, 2002): X ≤st Y ⇒ X ≤ttt Y ⇒ X ≤icv Y.
(1.1)
Applications, properties and interpretations of the T T T -transform order in the statistical theory of reliability and economic can be found in Kochar et al. (2002), Li and Zuo (2004) and Ahmad and Kayid (2004). In the context of lifetime distributions, some of the above orderings of distributions have been used to give characterizations and new definitions of ageing classes. By ageing we mean the phenomenon whereby an older system has a shorter remaining lifetime, in some statistical sense than a younger one (Bryson and Siddiqui, 1969). 2
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One of the most important approaches to the study of ageing is based on the concept of the residual life. For any random variable X, let Xt = [X − t|X > t],
t ∈ {x : F (x) < 1},
denote a random variable whose distribution is the same as the conditional distribution of X − t given that X > t. When X is the lifetime of a device, Xt can be regarded as the residual lifetime of the device at time t, given that the device has survived up to time t. We say that X is (i) new better than used (denoted by X ∈ NBU ) if Xt ≤st X
for all t ≥ 0;
(ii) new better than used in the increasing concave order (denoted by X ∈ NBU(2)) if Xt ≤icv X for all t ≥ 0. The classes NBU and NBU (2) have proved to be very useful in performing analyses of life lengths as well as usable in replacement policies. Hence a lot of results related to these two classes have been obtained in the literature (see for instance, Bryson and Siddiqui (1969), Barlow and Proschan (1981), Deshpand et al. (1986), Li and Kochar (2001), Franco et al. (2001) and Hu and Xie. (2002)). It seems, however, that the NBU property is too strong to verify as an ageing property while the NBU(2) property is too weak to characterize ageing. Hence one is motivated to find an ageing class in between these two and that is precisely what we do in the current work with the NBUT class. Suppose that F is the distribution function of lifetime of a unit, which could be a living organism or a mechanical component or a system, to determine whether the component is ageing with time for the T T T -transform associated with this distribution we need to compare the lifetime of a component TX (p) with its residual life TXt (p) at different ages and hence another class corresponding to the T T T transform order is needed and that is what we do here. Precisely, we have: Definition 1.1. A random variable X or F is said to be new better than used in the total time on test transform order denoted by NBUT if Xt ≤ttt X, equivalently, X ∈ NBUT if and only if Z
0
Ft−1 (p)
F (u + t) du ≤ F (t)
Z
F −1 (p)
F (u) du for all p ∈ (0, 1).
0
3
(1.2)
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The relations between the above ageing classes are the following: NBU
=⇒ NBU T
=⇒ NBU(2).
In general, the converse of the above relation is not necessarily true. Example 3.2 in Kochar et al. (2002) shows that X ≤ttt Y
; X ≤st Y,
and hence we conclude that the NBUT ; NBU. Again, from the fact that the T T T -transform order is a stochastic order that combines comparison of location with comparison of variation together with Theorem 2.2. in Kochar et al. (2002) we also conclude that the NBU(2) ; NBUT. Hence our class is a nontrivial and practical ageing class in between the NBU and NBU(2) classes. Next, in Section 2 several other properties of the NBUT class are presented, including the preservation under some reliability operations. In Section 3, behavior of the NBUT ageing notion is developed in terms of the monotonicity of the residual life of k−out of−n systems given the time of the (n − k)-th failure. Also, in that section, similar conclusion based on the residual life of parallel systems is presented as well. Finally, in Section 4 we address the question of testing H0 : F is exponential against H1 : F ∈ NBUT and not exponential. Throughout the paper we will use the term increasing in place of non-decreasing, and decreasing in place of non-increasing. All integrals and expectations are implicitly assumed to exist whenever they are written.
2
Preservation properties
Useful properties of ageing classes of life distributions are the closure with respect to typical reliability operations (see, e.g., Barlow and Proschan, 1981). In this section we present some preservation results for the NBUT class under random minima, series systems and mixture. Let X1 , X2 , ... be a sequence of independent and identical distributed (i.i.d) random variables and N be a positive integer-valued random variable which is independent of the Xi . Put X(1:N ) ≡ min{X1 , X2 , ..., XN }, X(N:N) ≡ max{X1 , X2 , ..., XN }. The random variables X(1:N) and X(N :N) arise naturally in reliability theory as the lifetimes of a series and parallel systems, respectively, with the random number N of identical components with lifetimes X1 , X2 , ..., XN . In life-testing, if a random 4
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censoring is adopted, then the completely observed data constitute a sample of random size, say X1 , X2 , ..., XN , where N > 0 is a random variable of integer value. In actuarial science, the claims received by an insurer in a certain time interval should also be a sample of random size, and, X(N:N ) , denotes the largest claim amount of the period. Also X(1:N) arises naturally in survival analysis as the minimal survival time of a transplant operation, where N of them are defective and hence may cause death. Some authors have made efforts to investigate preservation properties of some stochastic orders under random minima and maxima while other have centered their attention on investigating behavior of ageing properties in coherent structure, parallel (series) systems, convolution, mixture and renewal process (see, Shaked (1975), Bartozewicz (2001), Li and Zuo (2004) and Ahmad and Kayid (2004)). In the following, we give a preservation result for the NBUT class under formation of random minima. First, we recall the following result about the preservation of the T T T -transform order (see Li and Zuo, 2004). Theorem 2.1. Let X1 , X2 , ... and Y1 , Y2 , ... each be a sequence of i.i.d. random variables, and N is independent of Xi ’s and Yi ’s. If Xi ’s and Yi ’s are both non-negative and with common left end point 0, then Xi ≤ttt Yi for i = 1, 2, ..., implies min{X1 , ..., XN } ≤ttt min{Y1 , ..., YN }. Next we give the result. Theorem 2.2. Let X1 , X2 , ... be a sequence of i.i.d. random lives, and N is independent of Xi ’s. If X1 is NBUT , then min{X1 , ..., XN } is also of NBU T property.
Proof.
X1 is NBUT , then, for all t ≥ 0, (Xi )t ≤ttt Xi By Theorem 2.1, we have, for all t ≥ 0, min {(X1 )t , ..., (XN )t } ≤ttt min {X1 , ..., XN } . Since, for any positive integer n, st
t ≥ 0,
st
t ≥ 0,
min {(X1 )t , ..., (Xn )t } = (min {X1 , ..., Xn })t , it holds also that min {(X1 )t , ..., (XN )t } = (min {X1 , ..., XN })t , 5
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and hence (min {X1 , ..., XN })t ≤ttt min {X1 , ..., XN } ,
t ≥ 0.
That is to say, min {X1 , ..., XN } is NBUT too.
||
On the other hand, it is a well known fact that some ageing notions are preserved under formation of parallel or series system (see Barlow and Proschan (1981), Abouammoh and El-Neweihi (1986), Hendi et al. (1993), Li and Kochar (2001) and Pellerey and Petakos (2002)). According to Theorem 2.2, the following corollary can be deduced as below. Corollary 2.1. Let X1 , X2 , ..., Xn be a set of NBUT independent identically distributed components and consider Tn = min{X1 , X2 , ..., Xn }. Then Tn ∈ NBU T.
Proof.
By (1.1) and Pellerey and Petakos (2002), we have that [Tn − t | Tn > t] ≤ttt min{{X1 − t|X1 > t} , ..., {Xn − t|Xn > t}}. And by Theorem 5.1(a) (Kochar et al, 2002), we get min{{X1 − t|X1 > t} , ..., {Xn − t|Xn > t}} ≤ttt Tn = min{X1 , X2 , ..., Xn }, ||
and then the result follows. Remark 2.1.
We point out that Corollary 2.1 is special case of Theorem 3.1 in the next section when k = 1. Finally, since mixtures of some exponential life distributions often belong to the decreasing failure rate (DF R) class (Barlow and Proschan, 1981), we conclude that the NBUT class is not closed under mixtures,
3
Behavior of NBUT of k-out-of-n systems
The k-out-of-n structure is a very popular type of redundancy in fault-tolerant systems. A lot of applications have been found in both industrial and military systems since its earlier appearance. The multi-engine system in an air plane, the multidisplay system in a cockpit and the multi-pump system in a hydraulic control system are all fault-tolerant systems. For example, in an air plane with 4 engines, it may be possible to fly the plane if only 2 engines are functioning. However, if less than 2 engines function, the plane will fail to fly. Thus, the air plane may be represented as a 2-out-of-4 system. It is tolerant of failures of up to 2 engines for minimal functioning of the plane. In practical situations, special attention is often paid to the additional 6
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time length that a air plane can continue to fly when 2 engines have been already in failure state. Consider a system of n elements with their random lives X1 , ..., Xn , respectively. The k-out of-n system consists of n independent and identically distributed components and works as long as at least k components are working; that is, it works if at most n − k components have failed. Thus, the life of a k-out-of-n system can be characterized by the (n − k + 1)-th order statistic Xn−k+1,n . In fact, a series system is an n-out-of-n system and a parallel system is a 1-out-of-n system. Given that the (n−k)-th failure has occurred, then the system will fail when the (n−k +1)-th failure occurs. Thus, in order to understand how the ageing property of the elements affect the ageing procedure of the total life of the whole system, it is of special interest to study the ageing procedure of the residual life after the (n − k)-th failure. The residual life of a k-out-of-n system given that the (n − k)-th failure occurs at time t ≥ 0, is represented by the following conditional random variable, RLSk,n,t = (Xn−k+1,n − Xn−k,n | Xn−k,n = t), and hence the total life of the (n − k + 1)-th failed element is LSk = Xn−k+1,n . In this way, using stochastic comparisons, some authors characterized some ageing distributions by the stochastic ordering of the residual life of the k-out-of-n system, given that the (n−k) th failure has occurred at different times. In particular, Langberg et al.(1980) presented the following characterizations, X is NBU
⇔ RLSk,n,t ≤st LSk ,
for all t ≥ 0 and for any integer k such that 1 ≤ k < n.
Afterward, Belzunce et al. (1999) and Li and Chen (2004) provided additional results on some other stochastic orders and ageing notions. Recently, Li and Zuo (2002) consider the residual life of the system with i.i.d. NBU (2) elements, they also pay special attention to the residual life of a 1-out-of-n (parallel) system given that the (n − k)-th failure occurs at time t ≥ 0, RLPk,n,t = (Xn,n − Xn−k,n |Xn−k,n = t),
and the life span of the longest one within those r components should be LPk = Xn,n , and get the following result, X is NBU(2) ⇔ RLSk,n,t ≤icv LSk , f or all t ≥ 0, 7
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Assume the system is composed of i.i.d. components. Then, RLSk,n,t = min{(X1 )t , ..., (Xk )t }, LSk = min{X1 , ..., Xk }, and RLPk,n,t = max{(X1 )t , ..., (Xk )t }, LPk = max{X1 , ..., Xk }, where (Xi )t , i = 1, ..., k are i.i.d. copies of Xt . Our main result of this section presents behaviors of NBU T class in terms of conditioned residual life. Theorem 3.1. Assume that X is continuous. (i) If X is NBU T , then LSk ≥ttt RLSk,n,t , for t ≥ 0 . (ii) If LPk ≥ttt RLPk,n,t , for t ≥ 0, then X is NBUT . Proof. Since Xi is NBUT , (Xi )t ≤ttt Xi
for all t ≥ 0, i = 1, 2, ...n.
(i) In view of Theorem 5.1 (a) (Kochar et al, 2002), it follows that min{(X1 )t , ..., (Xk )t } ≤ttt min{X1 , ..., Xr }. Thus, we arrive the desired result. (ii) Since the T T T order has the reversed preservation property under the taking of maximum of i.i.d. components (Theorem 3.3 (ii), Li and Yam (2004)), max{(X1 )t , ..., (Xk )t } ≤ttt max{X1 , ..., Xk }. Hence, X is NBUT.
4
Testing against NBUT alternatives
In reliability analysis, testing the ageless notion (the exponential distribution) against positive ageing has been wide spread and of interest for well over four decades. The NBU positive ageing has been discussed and tested early on through the work of Hollander and Proschan (1972) and was followed by many authors. For a recent literature review and new approaches to this problem we refer the readers to Ahmad 8
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(2001), Ahmad et al. (2001) and Ahmad and Mugdadi (2004), where other classes of positive ageing are also tested and compared. Since the NBUT class we presented here includes the NBU as a subclass and hence is an easier to verify ageing property it would be of interest to test H0 : F is exponential against the alternative H1 : F is NBU T and not exponential. If we use the approach taken by Hollander and Proschan (1972) it is possible that we obtain a test procedure that is weaker(in the sense of Pitman efficiency). We however, take another new approach that yields better testing in our case than that of the NBU one even though our class is much bigger. First, we need a measure of departure from H0 in favor of H1 . The following lemma leads to such measure. Lemma 4.1. For X1 , X2 and X3 which are i.i.d. and NBUT , 2 E {min (X1 , X2 , X3 )} ≥ E {min (X1 , X2 )} . 3
(4.1)
Proof. First observe that if X1 , ..., Xk are independent random variables with same distribution function F (·) and survival function F (·), then it is easy to see that Z ∞ k E {min (X1 , .., Xk )} = F (u)du. 0
According to (1.2), we have that: Z 1Z 0
∞
2
F (t)dF (t)
Z
F −1 (p)
0
0
Z 1Z F (x) dxdp− 0
∞
F (t)
0
Z
0
Ft−1 (p)
F (x+t) dxdF (t)dp ≥ 0.
But Z 1Z 0
∞
Z
2
F (t)dF (t)
F −1 (p)
F (x) dxdp
0
0
Z Z −1 Z Z 1 ∞ u 1 1 F (p) F (x) dxdp = F (x) dF (u)dx = 3 0 0 3 0 0 Z 1 1 ∞ 2 F (x)dx = E {min (X1 , X2 )} . = (4.2) 3 0 3
Next, Z 1Z 0
0
∞
F (t)
Z
Ft−1 (p)
F (x + t) dxdF (t)dp
0
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Z
=
0
=− Z
=
0
∞Z ∞
F (t)
0
Z
0
Now,
R∞ x
u
F (x + t) dxdFt (u)dF (t)
0
∞ Z ∞ Z u+t 0
∞Z ∞ 0
F (w) dwdF (u + t)dF (t)
t
[υ(t) − υ(u + t)] du F (u + t)dF (t)
def
where υ(x) =
Z
≡ I − II,
say,
F (u)du.
I =
Z
0
= −
∞
Z
Z
∞
υ(t)du F (u + t)dF (t)
0
∞
υ(t)F (t)dF (t).
0
Also, II =
Z
0
= − = −
∞Z ∞
Z
∞
0
Z
υ(u + t)du F (u + t)dF (t)
0
∞
0
Hence I − II =
υ(t)F (t)dF (t) − υ(t)F (t)dF (t) − Z
0
∞
2
F (t)dt −
Z
Z
∞
2
F (t)F (t)dt
0
Z
∞
2
F (t)dt +
0
∞
Z
∞
3
F (t)dt.
0
3
F (t)dt
0
= E {min(X1 , X2 )} − E {min(X1 , X2 , X3 )} . The result follows form (4.2) and (4.3). The measure of departure from H0 in favor of H1 maybe taken to be δ = E{min(X1 , X2 , X3 ) − (2/3) min(X1 , X2 )}.
(4.3) || (4.4)
Note that under H0 , δ = 0 while it is positive under H1 . To make the test scale invariant we take ∆ = δ/µ . Let X1 , ..., Xn be a random sample from F . We estimate µ by X and δ by: ½ ¾ f 1 X X X 2 min(Xi , Xj , Xk ) − min(Xi , Xj ) , δ= (4.5) 6=j 6=k i (n)3 3 10
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where (n)3 = n(n − 1)(n − 2).
Using standard U-statistics theory we can easily prove the following:
Theorem 4.1.
¶ µ f √ As n → ∞, n ∆ −∆ is asymptotically normal with 0 mean and variance
σ 2 = µ−2 V (Ψ1 (X1 )) where Z X1 2 Ψ1 (X1 ) = 6 xF (x)dF (x) + 3X1 F (X1 ) 0 Z Z 4 4 X1 2 ∞ 2 − X1 F (X1 ) − xdF (x) − F (x)dx. (4.6) 3 3 0 3 0 Under H0 , σ 20 =
2 . 135
Proof. f
All we need to calculate is asymptotic variance of δ . Set ϕ (X1 , X2 , X3 ) = min (X1 , X2 , X3 ) −
2 min (X1 , X2 ) . 3
Hence we easily see that ϕ1 (X1 ) = E [φ(X1 , X2 , X3 ) | X1 ] = ϕ2 (X1 ) = [φ(X2 , X1 , X3 ) | X1 ] = 2
Z
0
X1
1 xF (x)dF (x) + X1 F (X1 ) − 3 2
Z
X1
0
2 xdF (x) − X1 F (X1 ). 3
Finally, ϕ3 (X1 ) = E [ϕ (X2 , X3 , X1 |X1 )] = 2
Z
0
X1
2 xF (x)dx + X1 F (X1 ) − 3
Result follows by setting Ψ (X1 ) =
2
X3
i=1
Z
∞
2
F (x)dx.
0
ϕi (X1 ).
|| Under H0 , direct calculations give the result. q f To carry out the test, calculate 135 and reject if this is larger than Zα the ∆ 2 standard normal variate. To assess the goodness of the above procedure we can use the concept of Pitman Asymptotic Efficacy (PAE ) defined as: ¯ ¯ ¯d ¯ ¯ P AE(∆) = P EA(δ) = ¯ δ θ ¯¯ /σ 0 , (4.7) dθ θ→θ0 11
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R 2 R 3 where δ θ = F θ − 23 F θ and θ0 is the null value of θ. Let us consider the following three distributions who are in the NBUT class since they are in the NBU class. θ
(1) The Weibull Distribution: F θ (x) = e−x
θ
2
(2) The Linear Failure Rate Distribution: F θ (x) = e−x− 2 x (3) The Makeham Distribution: F θ (x) = e−x−θ(e
).
−x +x−1
Note that the exponential is at θ = 1, 0 and θ, respectively. Calculating the PAEs of the above alternatives we get the values: 1.1104 for the Weibull, 0.5705 for the linear failure rate and 0.288 for the Makeham, respectively. Note also that the NBU test of Hollander and Proschan (1972) has efficacy values of the above three alternatives equal to 1.1619, 0.5095 and 0.2582, respectively. Thus our test here is better for the linear failure rate and for the Makeham while slightly worse for the Wiebull even though our class is much larger than that of the NBU. Finally, we point out that when H0 is rejected and thus F is NBUT , estimating of F as well as other related functions such as the hazard rate or the mean residual life time is an interesting and open question. The authors are now working on these and related questions and hope to have results soon. The main thrust of the work of estimating an NBUT distribution is to notice that (Z −1 ) Z −1 Ft
∗
F (t) = sup
Ft
F (x + t)dx /
0≤p≤1
0
(4.8)
F (x)dx ,
0
is NBU T life distribution. µ −1 ¶ f −1 f Thus, if F t (p) F (p) is non-parametric estimate of Ft−1 (p) (F −1 (p)) then the estimate
∗
F n (t) = sup
0≤p≤1
Z
f −1
Ft
F n (x + t)dx /
0
Z
0
f −1
Ft
F n (x)dx ,
(4.9) ∗
is NBUT . We shall prove the above results and study the properties of F n (t) in a subsequent work.
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ACKNOWLEDGMENT The authors are indebted to the Editor and two referees for their careful and detailed remarks which helped improve both content and presentation of this work. (M. K.) would like to thank Professor Franco Pellerey for some useful comments and suggestions regarding the first draft of this work. REFERENCES Abouammoh, A. and El-Neweihi, E. (1986). closure of the NBUE and DMRL classes under formation of parallel system. Statistics & Probability Letters, 4, 223-225. Ahmad, I. A. (2001). Moments inequalities of aging families of distributions with hypothesis testing applications. Journal of Statistical Planning and Inference, 92, 121-132. Ahmad, I. A., Al-Wasel, I. A. and Mugdadi, A. R. (2001). A goodness of fit approach to major life testing problems. International Journal of Reliability and Applications, 2, 81-98. Ahmad, I. A. and Mugdadi, A. R. (2004). Further moment inequalities of life distributions with hypothesis testing applications: the IFRA, NBUC, DMRL classes. Journal of Statistical Planning and Inference, 120, 1-12. Ahmad, I. A. and Kayid, M. (2004). Reversed preservation of stochastic orders for random minima and maxima, with applications. submitted. Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. To Begin with. Silver Spring, M D. Belzunce, F., Franco, M and Ruiz, J. M. (1999). On aging properties based on the residual life of k-out of-n system. Probability in the Engineering and Informational Sciences, 13, 193-199. Bryson, M. C. and Siddiqui, M. M. (1969). Some criteria for aging. Journal of the American Statistical Association, 64, 1472-1483. Bartoszewicz, J. (2001). Stochastic comparisons of random minima and maxima from life distributions. Statistics & Probability Letters, 55, 107-112. Cao, J. and Wang, Y. (1991). The NBUC and NWUC classes of life distributions. Journal of Applied Probability, 28, 473-479. Deshpande, J. V., Kochar, S. C. and Singh, H. (1986). Aspects of positive aging. Journal of Applied Probability, 23, 748-758. Franco, M., Ruiz, J. M and Ruiz, M. C. (2001). On closure of the IF R(2) and NBU (2) classes. Journal of Applied Probability, 38, 236-242. 13
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Hollander, M. and Proschan, F.(1972). Testing whether new is better than used. Annals of Mathematical Statistics, 43, 1136-1146. Hendi, M. and Mashhour, A. (1993). Closure of the NBUC class under formation of parallel systems. Journal of Applied Probability, 31, 458-469. Hu, T. and Xie, H. (2002). Proofs of the closure property of NBUC and NBU(2) under convolution. Journal of Applied Probability, 39, 224-227. Kochar, S. C., Li, X. and Shaked, M. (2002). The total time on test transform and the excess wealth stochastic orders of distributions. Advances in Applied Probability. 34, 826-845. Langberg, N. A., Leon, R. V. and Proschan, F. (1980). Characterizations of nonparametric classes of life distributions. The Annals of Probability, 8, 1163-1170. Li, X. and Kochar, S. C. (2001). Some new results of NBU (2) class of life distributions. Journal of Applied Probability, 38, 237-242. Li, X. and Zuo, M. (2002). On behaviors of some new aging properties based upon the residual life of k-out of-n systems. Journal of Applied Probability, 39, 426-433. Li, X. and Yam, R. C. M. (2004). Reversed preservation properties of some negative aging conceptions. Statistical Papers, to appear. Li, X. and Zuo, M. (2004). Preservation of stochastic orders for random minima and maxima, with applications. Naval Research Logistics, to appear. Li, X. and Chen, J. (2004). Aging properties of the residual life length of k-out-of-n systems with independent but non-identical components. Applied Stochastic Models in Business and Industry, to appear. Loh, W. Y. (1984). A new generalization of the class of NBU distribution. IEEE Transaction on Reliability, 33, 419-422. Muller, A. and Stoyan, D. (2002). Comparison methods for queues and other stochastic models. Wiley & Sons, New York, NY. Marshall, A.W. and Proschan, F. (1972). Classes of life distributions applicable in replacement with renewal theory implications. Sixth Berkeley Smyp. Mathematical Statistics & Probability, 1, 395-415. Pellerey, F. and Petakos, K (2002). On closure property of the NBU C class under formation of parallel systems. IEEE Transaction on Reliability, 51, 452-454. Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, New York. Shaked, M. (1975). On the distribution of the minimum and of the maximum of a random number of i.i.d. random variables. In: Patil, G.P., Kotz, S., Ord, J.(Eds.), Statistical Distributions in Scientific Work, Vol. 1. Reidel, Dordrecht, 363-380. 14
THE NBUT CLASS OF LIFE DISTRIBUTIONS I. A. Ahmad∗, M. Kayid †and X. Li‡ IEEE Transaction on Reliability, to appear, June, 2005
Abstract A new class of life distributions, namely new better than used in the total time on test transform ordering (N BU T ) is introduced. The relations of this class to other classes of life distributions, closure properties under some reliability operations are discussed as well. We provide a simple argument based on stochastic orders that the family of the N BU T distributions is closed under the formation of series systems in case of independent identically distributed components. Behavior of this class is developed in terms of the monotonicity of the residual life of k−out of−n systems given the time of the (n − k)-th failure. Finally, testing exponentially against N BU T class is discussed
Keywords: T T T transform order, stochastic order, increasing concave order, random minima, series system, mixing, k out-of-n systems, life testing
1
Introduction and motivation
Since the accurate distribution of the life of an element or a system is often unavailable in practical situation, nonparametric ageing properties have been found to be quite useful in modeling ageing or wear-out process and to conduct maintenance policy in reliability. Such ageing classes are derived via several notions of comparison between random variables. Of the most commonly used comparison we find, cf. Muller and Sotayan (2002) and Shaked and Shanthikumar (1994), the stochastic comparison and the increasing concave comparison. Formally, if X and Y are two random variables with distributions F and G (survivals F and G), respectively, then we say that X is smaller than Y in the: ∗
Dept. of Stat. and Act. Science, University of Central Fl., Orlando, Florida 32816-2370, USA Dept. of Mathematics, Faculty of Education (Suez ), Suez Canal University, EGYPT ‡ Dept. of Mathematics, Lanzhou University, Lanzhou 730000, People’s Republic of CHINA †
1
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(i) stochastic order (denoted by X ≤st Y ), if E [φ(X)] ≤ E [φ(Y )]
for all increasing functions φ;
(ii) increasing concave order (denoted by X ≤icv Y ), if E[φ(X)] ≤ E[φ(Y )] for all increasing concave functions φ. Consider a distribution function F , of a non-negative random variable X, which is strictly increasing on its interval support. Let p ∈ (0, 1) and t ≥ 0 be two values related by t = F −1 (p), where F −1 is the right continuous inverse of F. Denote by ª © AF ≡ (x, u) : u ∈ (0, p), x ∈ (0, F −1 (u)) , and
ª © BF ≡ (x, u) : u ∈ (p, 1), x ∈ (0, F −1 (u)) .
The above areas of the regions have various intuitive meanings in different applications. For example, if F is the distribution function of the lifetime of a machine, then TX (p) ≡ kAF (p) ∪ BF (p)k , p ∈ (0, 1), corresponds to the total time on test (T T T ) transform associated with this distribution. Recently the total time on test (T T T ) transform order has also come to use in reliability and life testing (see, Kochar et al. (2002)). We say that a random variable X is smaller than a random variable Y in the T T T -transform order (denoted by X ≤ttt Y ) if, and only if, TX (p) =
Z
0
F −1 (p)
F (x) dx ≤
Z
G−1 (p)
G(x) dx = TY (p),
0
p ∈ (0, 1),
where F −1 and G−1 are respectively the right continuous inverses of F and G. The T T T -transform order is weaker than the stochastic order (≤st ) but is stronger than the increasing concave order (≤icv ). For the previous orders we have the following relationships (see, Kochar et al, 2002): X ≤st Y ⇒ X ≤ttt Y ⇒ X ≤icv Y.
(1.1)
Applications, properties and interpretations of the T T T -transform order in the statistical theory of reliability and economic can be found in Kochar et al. (2002), Li and Zuo (2004) and Ahmad and Kayid (2004). In the context of lifetime distributions, some of the above orderings of distributions have been used to give characterizations and new definitions of ageing classes. By ageing we mean the phenomenon whereby an older system has a shorter remaining lifetime, in some statistical sense than a younger one (Bryson and Siddiqui, 1969). 2
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One of the most important approaches to the study of ageing is based on the concept of the residual life. For any random variable X, let Xt = [X − t|X > t],
t ∈ {x : F (x) < 1},
denote a random variable whose distribution is the same as the conditional distribution of X − t given that X > t. When X is the lifetime of a device, Xt can be regarded as the residual lifetime of the device at time t, given that the device has survived up to time t. We say that X is (i) new better than used (denoted by X ∈ NBU ) if Xt ≤st X
for all t ≥ 0;
(ii) new better than used in the increasing concave order (denoted by X ∈ NBU(2)) if Xt ≤icv X for all t ≥ 0. The classes NBU and NBU (2) have proved to be very useful in performing analyses of life lengths as well as usable in replacement policies. Hence a lot of results related to these two classes have been obtained in the literature (see for instance, Bryson and Siddiqui (1969), Barlow and Proschan (1981), Deshpand et al. (1986), Li and Kochar (2001), Franco et al. (2001) and Hu and Xie. (2002)). It seems, however, that the NBU property is too strong to verify as an ageing property while the NBU(2) property is too weak to characterize ageing. Hence one is motivated to find an ageing class in between these two and that is precisely what we do in the current work with the NBUT class. Suppose that F is the distribution function of lifetime of a unit, which could be a living organism or a mechanical component or a system, to determine whether the component is ageing with time for the T T T -transform associated with this distribution we need to compare the lifetime of a component TX (p) with its residual life TXt (p) at different ages and hence another class corresponding to the T T T transform order is needed and that is what we do here. Precisely, we have: Definition 1.1. A random variable X or F is said to be new better than used in the total time on test transform order denoted by NBUT if Xt ≤ttt X, equivalently, X ∈ NBUT if and only if Z
0
Ft−1 (p)
F (u + t) du ≤ F (t)
Z
F −1 (p)
F (u) du for all p ∈ (0, 1).
0
3
(1.2)
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The relations between the above ageing classes are the following: NBU
=⇒ NBU T
=⇒ NBU(2).
In general, the converse of the above relation is not necessarily true. Example 3.2 in Kochar et al. (2002) shows that X ≤ttt Y
; X ≤st Y,
and hence we conclude that the NBUT ; NBU. Again, from the fact that the T T T -transform order is a stochastic order that combines comparison of location with comparison of variation together with Theorem 2.2. in Kochar et al. (2002) we also conclude that the NBU(2) ; NBUT. Hence our class is a nontrivial and practical ageing class in between the NBU and NBU(2) classes. Next, in Section 2 several other properties of the NBUT class are presented, including the preservation under some reliability operations. In Section 3, behavior of the NBUT ageing notion is developed in terms of the monotonicity of the residual life of k−out of−n systems given the time of the (n − k)-th failure. Also, in that section, similar conclusion based on the residual life of parallel systems is presented as well. Finally, in Section 4 we address the question of testing H0 : F is exponential against H1 : F ∈ NBUT and not exponential. Throughout the paper we will use the term increasing in place of non-decreasing, and decreasing in place of non-increasing. All integrals and expectations are implicitly assumed to exist whenever they are written.
2
Preservation properties
Useful properties of ageing classes of life distributions are the closure with respect to typical reliability operations (see, e.g., Barlow and Proschan, 1981). In this section we present some preservation results for the NBUT class under random minima, series systems and mixture. Let X1 , X2 , ... be a sequence of independent and identical distributed (i.i.d) random variables and N be a positive integer-valued random variable which is independent of the Xi . Put X(1:N ) ≡ min{X1 , X2 , ..., XN }, X(N:N) ≡ max{X1 , X2 , ..., XN }. The random variables X(1:N) and X(N :N) arise naturally in reliability theory as the lifetimes of a series and parallel systems, respectively, with the random number N of identical components with lifetimes X1 , X2 , ..., XN . In life-testing, if a random 4
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censoring is adopted, then the completely observed data constitute a sample of random size, say X1 , X2 , ..., XN , where N > 0 is a random variable of integer value. In actuarial science, the claims received by an insurer in a certain time interval should also be a sample of random size, and, X(N:N ) , denotes the largest claim amount of the period. Also X(1:N) arises naturally in survival analysis as the minimal survival time of a transplant operation, where N of them are defective and hence may cause death. Some authors have made efforts to investigate preservation properties of some stochastic orders under random minima and maxima while other have centered their attention on investigating behavior of ageing properties in coherent structure, parallel (series) systems, convolution, mixture and renewal process (see, Shaked (1975), Bartozewicz (2001), Li and Zuo (2004) and Ahmad and Kayid (2004)). In the following, we give a preservation result for the NBUT class under formation of random minima. First, we recall the following result about the preservation of the T T T -transform order (see Li and Zuo, 2004). Theorem 2.1. Let X1 , X2 , ... and Y1 , Y2 , ... each be a sequence of i.i.d. random variables, and N is independent of Xi ’s and Yi ’s. If Xi ’s and Yi ’s are both non-negative and with common left end point 0, then Xi ≤ttt Yi for i = 1, 2, ..., implies min{X1 , ..., XN } ≤ttt min{Y1 , ..., YN }. Next we give the result. Theorem 2.2. Let X1 , X2 , ... be a sequence of i.i.d. random lives, and N is independent of Xi ’s. If X1 is NBUT , then min{X1 , ..., XN } is also of NBU T property.
Proof.
X1 is NBUT , then, for all t ≥ 0, (Xi )t ≤ttt Xi By Theorem 2.1, we have, for all t ≥ 0, min {(X1 )t , ..., (XN )t } ≤ttt min {X1 , ..., XN } . Since, for any positive integer n, st
t ≥ 0,
st
t ≥ 0,
min {(X1 )t , ..., (Xn )t } = (min {X1 , ..., Xn })t , it holds also that min {(X1 )t , ..., (XN )t } = (min {X1 , ..., XN })t , 5
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and hence (min {X1 , ..., XN })t ≤ttt min {X1 , ..., XN } ,
t ≥ 0.
That is to say, min {X1 , ..., XN } is NBUT too.
||
On the other hand, it is a well known fact that some ageing notions are preserved under formation of parallel or series system (see Barlow and Proschan (1981), Abouammoh and El-Neweihi (1986), Hendi et al. (1993), Li and Kochar (2001) and Pellerey and Petakos (2002)). According to Theorem 2.2, the following corollary can be deduced as below. Corollary 2.1. Let X1 , X2 , ..., Xn be a set of NBUT independent identically distributed components and consider Tn = min{X1 , X2 , ..., Xn }. Then Tn ∈ NBU T.
Proof.
By (1.1) and Pellerey and Petakos (2002), we have that [Tn − t | Tn > t] ≤ttt min{{X1 − t|X1 > t} , ..., {Xn − t|Xn > t}}. And by Theorem 5.1(a) (Kochar et al, 2002), we get min{{X1 − t|X1 > t} , ..., {Xn − t|Xn > t}} ≤ttt Tn = min{X1 , X2 , ..., Xn }, ||
and then the result follows. Remark 2.1.
We point out that Corollary 2.1 is special case of Theorem 3.1 in the next section when k = 1. Finally, since mixtures of some exponential life distributions often belong to the decreasing failure rate (DF R) class (Barlow and Proschan, 1981), we conclude that the NBUT class is not closed under mixtures,
3
Behavior of NBUT of k-out-of-n systems
The k-out-of-n structure is a very popular type of redundancy in fault-tolerant systems. A lot of applications have been found in both industrial and military systems since its earlier appearance. The multi-engine system in an air plane, the multidisplay system in a cockpit and the multi-pump system in a hydraulic control system are all fault-tolerant systems. For example, in an air plane with 4 engines, it may be possible to fly the plane if only 2 engines are functioning. However, if less than 2 engines function, the plane will fail to fly. Thus, the air plane may be represented as a 2-out-of-4 system. It is tolerant of failures of up to 2 engines for minimal functioning of the plane. In practical situations, special attention is often paid to the additional 6
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time length that a air plane can continue to fly when 2 engines have been already in failure state. Consider a system of n elements with their random lives X1 , ..., Xn , respectively. The k-out of-n system consists of n independent and identically distributed components and works as long as at least k components are working; that is, it works if at most n − k components have failed. Thus, the life of a k-out-of-n system can be characterized by the (n − k + 1)-th order statistic Xn−k+1,n . In fact, a series system is an n-out-of-n system and a parallel system is a 1-out-of-n system. Given that the (n−k)-th failure has occurred, then the system will fail when the (n−k +1)-th failure occurs. Thus, in order to understand how the ageing property of the elements affect the ageing procedure of the total life of the whole system, it is of special interest to study the ageing procedure of the residual life after the (n − k)-th failure. The residual life of a k-out-of-n system given that the (n − k)-th failure occurs at time t ≥ 0, is represented by the following conditional random variable, RLSk,n,t = (Xn−k+1,n − Xn−k,n | Xn−k,n = t), and hence the total life of the (n − k + 1)-th failed element is LSk = Xn−k+1,n . In this way, using stochastic comparisons, some authors characterized some ageing distributions by the stochastic ordering of the residual life of the k-out-of-n system, given that the (n−k) th failure has occurred at different times. In particular, Langberg et al.(1980) presented the following characterizations, X is NBU
⇔ RLSk,n,t ≤st LSk ,
for all t ≥ 0 and for any integer k such that 1 ≤ k < n.
Afterward, Belzunce et al. (1999) and Li and Chen (2004) provided additional results on some other stochastic orders and ageing notions. Recently, Li and Zuo (2002) consider the residual life of the system with i.i.d. NBU (2) elements, they also pay special attention to the residual life of a 1-out-of-n (parallel) system given that the (n − k)-th failure occurs at time t ≥ 0, RLPk,n,t = (Xn,n − Xn−k,n |Xn−k,n = t),
and the life span of the longest one within those r components should be LPk = Xn,n , and get the following result, X is NBU(2) ⇔ RLSk,n,t ≤icv LSk , f or all t ≥ 0, 7
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Assume the system is composed of i.i.d. components. Then, RLSk,n,t = min{(X1 )t , ..., (Xk )t }, LSk = min{X1 , ..., Xk }, and RLPk,n,t = max{(X1 )t , ..., (Xk )t }, LPk = max{X1 , ..., Xk }, where (Xi )t , i = 1, ..., k are i.i.d. copies of Xt . Our main result of this section presents behaviors of NBU T class in terms of conditioned residual life. Theorem 3.1. Assume that X is continuous. (i) If X is NBU T , then LSk ≥ttt RLSk,n,t , for t ≥ 0 . (ii) If LPk ≥ttt RLPk,n,t , for t ≥ 0, then X is NBUT . Proof. Since Xi is NBUT , (Xi )t ≤ttt Xi
for all t ≥ 0, i = 1, 2, ...n.
(i) In view of Theorem 5.1 (a) (Kochar et al, 2002), it follows that min{(X1 )t , ..., (Xk )t } ≤ttt min{X1 , ..., Xr }. Thus, we arrive the desired result. (ii) Since the T T T order has the reversed preservation property under the taking of maximum of i.i.d. components (Theorem 3.3 (ii), Li and Yam (2004)), max{(X1 )t , ..., (Xk )t } ≤ttt max{X1 , ..., Xk }. Hence, X is NBUT.
4
Testing against NBUT alternatives
In reliability analysis, testing the ageless notion (the exponential distribution) against positive ageing has been wide spread and of interest for well over four decades. The NBU positive ageing has been discussed and tested early on through the work of Hollander and Proschan (1972) and was followed by many authors. For a recent literature review and new approaches to this problem we refer the readers to Ahmad 8
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(2001), Ahmad et al. (2001) and Ahmad and Mugdadi (2004), where other classes of positive ageing are also tested and compared. Since the NBUT class we presented here includes the NBU as a subclass and hence is an easier to verify ageing property it would be of interest to test H0 : F is exponential against the alternative H1 : F is NBU T and not exponential. If we use the approach taken by Hollander and Proschan (1972) it is possible that we obtain a test procedure that is weaker(in the sense of Pitman efficiency). We however, take another new approach that yields better testing in our case than that of the NBU one even though our class is much bigger. First, we need a measure of departure from H0 in favor of H1 . The following lemma leads to such measure. Lemma 4.1. For X1 , X2 and X3 which are i.i.d. and NBUT , 2 E {min (X1 , X2 , X3 )} ≥ E {min (X1 , X2 )} . 3
(4.1)
Proof. First observe that if X1 , ..., Xk are independent random variables with same distribution function F (·) and survival function F (·), then it is easy to see that Z ∞ k E {min (X1 , .., Xk )} = F (u)du. 0
According to (1.2), we have that: Z 1Z 0
∞
2
F (t)dF (t)
Z
F −1 (p)
0
0
Z 1Z F (x) dxdp− 0
∞
F (t)
0
Z
0
Ft−1 (p)
F (x+t) dxdF (t)dp ≥ 0.
But Z 1Z 0
∞
Z
2
F (t)dF (t)
F −1 (p)
F (x) dxdp
0
0
Z Z −1 Z Z 1 ∞ u 1 1 F (p) F (x) dxdp = F (x) dF (u)dx = 3 0 0 3 0 0 Z 1 1 ∞ 2 F (x)dx = E {min (X1 , X2 )} . = (4.2) 3 0 3
Next, Z 1Z 0
0
∞
F (t)
Z
Ft−1 (p)
F (x + t) dxdF (t)dp
0
9
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Z
=
0
=− Z
=
0
∞Z ∞
F (t)
0
Z
0
Now,
R∞ x
u
F (x + t) dxdFt (u)dF (t)
0
∞ Z ∞ Z u+t 0
∞Z ∞ 0
F (w) dwdF (u + t)dF (t)
t
[υ(t) − υ(u + t)] du F (u + t)dF (t)
def
where υ(x) =
Z
≡ I − II,
say,
F (u)du.
I =
Z
0
= −
∞
Z
Z
∞
υ(t)du F (u + t)dF (t)
0
∞
υ(t)F (t)dF (t).
0
Also, II =
Z
0
= − = −
∞Z ∞
Z
∞
0
Z
υ(u + t)du F (u + t)dF (t)
0
∞
0
Hence I − II =
υ(t)F (t)dF (t) − υ(t)F (t)dF (t) − Z
0
∞
2
F (t)dt −
Z
Z
∞
2
F (t)F (t)dt
0
Z
∞
2
F (t)dt +
0
∞
Z
∞
3
F (t)dt.
0
3
F (t)dt
0
= E {min(X1 , X2 )} − E {min(X1 , X2 , X3 )} . The result follows form (4.2) and (4.3). The measure of departure from H0 in favor of H1 maybe taken to be δ = E{min(X1 , X2 , X3 ) − (2/3) min(X1 , X2 )}.
(4.3) || (4.4)
Note that under H0 , δ = 0 while it is positive under H1 . To make the test scale invariant we take ∆ = δ/µ . Let X1 , ..., Xn be a random sample from F . We estimate µ by X and δ by: ½ ¾ f 1 X X X 2 min(Xi , Xj , Xk ) − min(Xi , Xj ) , δ= (4.5) 6=j 6=k i (n)3 3 10
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where (n)3 = n(n − 1)(n − 2).
Using standard U-statistics theory we can easily prove the following:
Theorem 4.1.
¶ µ f √ As n → ∞, n ∆ −∆ is asymptotically normal with 0 mean and variance
σ 2 = µ−2 V (Ψ1 (X1 )) where Z X1 2 Ψ1 (X1 ) = 6 xF (x)dF (x) + 3X1 F (X1 ) 0 Z Z 4 4 X1 2 ∞ 2 − X1 F (X1 ) − xdF (x) − F (x)dx. (4.6) 3 3 0 3 0 Under H0 , σ 20 =
2 . 135
Proof. f
All we need to calculate is asymptotic variance of δ . Set ϕ (X1 , X2 , X3 ) = min (X1 , X2 , X3 ) −
2 min (X1 , X2 ) . 3
Hence we easily see that ϕ1 (X1 ) = E [φ(X1 , X2 , X3 ) | X1 ] = ϕ2 (X1 ) = [φ(X2 , X1 , X3 ) | X1 ] = 2
Z
0
X1
1 xF (x)dF (x) + X1 F (X1 ) − 3 2
Z
X1
0
2 xdF (x) − X1 F (X1 ). 3
Finally, ϕ3 (X1 ) = E [ϕ (X2 , X3 , X1 |X1 )] = 2
Z
0
X1
2 xF (x)dx + X1 F (X1 ) − 3
Result follows by setting Ψ (X1 ) =
2
X3
i=1
Z
∞
2
F (x)dx.
0
ϕi (X1 ).
|| Under H0 , direct calculations give the result. q f To carry out the test, calculate 135 and reject if this is larger than Zα the ∆ 2 standard normal variate. To assess the goodness of the above procedure we can use the concept of Pitman Asymptotic Efficacy (PAE ) defined as: ¯ ¯ ¯d ¯ ¯ P AE(∆) = P EA(δ) = ¯ δ θ ¯¯ /σ 0 , (4.7) dθ θ→θ0 11
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R 2 R 3 where δ θ = F θ − 23 F θ and θ0 is the null value of θ. Let us consider the following three distributions who are in the NBUT class since they are in the NBU class. θ
(1) The Weibull Distribution: F θ (x) = e−x
θ
2
(2) The Linear Failure Rate Distribution: F θ (x) = e−x− 2 x (3) The Makeham Distribution: F θ (x) = e−x−θ(e
).
−x +x−1
Note that the exponential is at θ = 1, 0 and θ, respectively. Calculating the PAEs of the above alternatives we get the values: 1.1104 for the Weibull, 0.5705 for the linear failure rate and 0.288 for the Makeham, respectively. Note also that the NBU test of Hollander and Proschan (1972) has efficacy values of the above three alternatives equal to 1.1619, 0.5095 and 0.2582, respectively. Thus our test here is better for the linear failure rate and for the Makeham while slightly worse for the Wiebull even though our class is much larger than that of the NBU. Finally, we point out that when H0 is rejected and thus F is NBUT , estimating of F as well as other related functions such as the hazard rate or the mean residual life time is an interesting and open question. The authors are now working on these and related questions and hope to have results soon. The main thrust of the work of estimating an NBUT distribution is to notice that (Z −1 ) Z −1 Ft
∗
F (t) = sup
Ft
F (x + t)dx /
0≤p≤1
0
(4.8)
F (x)dx ,
0
is NBU T life distribution. µ −1 ¶ f −1 f Thus, if F t (p) F (p) is non-parametric estimate of Ft−1 (p) (F −1 (p)) then the estimate
∗
F n (t) = sup
0≤p≤1
Z
f −1
Ft
F n (x + t)dx /
0
Z
0
f −1
Ft
F n (x)dx ,
(4.9) ∗
is NBUT . We shall prove the above results and study the properties of F n (t) in a subsequent work.
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ACKNOWLEDGMENT The authors are indebted to the Editor and two referees for their careful and detailed remarks which helped improve both content and presentation of this work. (M. K.) would like to thank Professor Franco Pellerey for some useful comments and suggestions regarding the first draft of this work. REFERENCES Abouammoh, A. and El-Neweihi, E. (1986). closure of the NBUE and DMRL classes under formation of parallel system. Statistics & Probability Letters, 4, 223-225. Ahmad, I. A. (2001). Moments inequalities of aging families of distributions with hypothesis testing applications. Journal of Statistical Planning and Inference, 92, 121-132. Ahmad, I. A., Al-Wasel, I. A. and Mugdadi, A. R. (2001). A goodness of fit approach to major life testing problems. International Journal of Reliability and Applications, 2, 81-98. Ahmad, I. A. and Mugdadi, A. R. (2004). Further moment inequalities of life distributions with hypothesis testing applications: the IFRA, NBUC, DMRL classes. Journal of Statistical Planning and Inference, 120, 1-12. Ahmad, I. A. and Kayid, M. (2004). Reversed preservation of stochastic orders for random minima and maxima, with applications. submitted. Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. To Begin with. Silver Spring, M D. Belzunce, F., Franco, M and Ruiz, J. M. (1999). On aging properties based on the residual life of k-out of-n system. Probability in the Engineering and Informational Sciences, 13, 193-199. Bryson, M. C. and Siddiqui, M. M. (1969). Some criteria for aging. Journal of the American Statistical Association, 64, 1472-1483. Bartoszewicz, J. (2001). Stochastic comparisons of random minima and maxima from life distributions. Statistics & Probability Letters, 55, 107-112. Cao, J. and Wang, Y. (1991). The NBUC and NWUC classes of life distributions. Journal of Applied Probability, 28, 473-479. Deshpande, J. V., Kochar, S. C. and Singh, H. (1986). Aspects of positive aging. Journal of Applied Probability, 23, 748-758. Franco, M., Ruiz, J. M and Ruiz, M. C. (2001). On closure of the IF R(2) and NBU (2) classes. Journal of Applied Probability, 38, 236-242. 13
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