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ON THE MEAN INACTIVITY TIME ORDERING WITH RELIABILITY APPLICATIONS
Published online by Cambridge University Press: 01 July 2004
Abstract
The purpose of this article is to study several preservation properties of stochastic comparisons based on the mean inactivity time order under the reliability operations of convolution and mixture. Characterizations and relationships with the other well-known orders are given. Some examples of interest in reliability theory are also presented. Finally, testing in the increasing mean inactivity time class is discussed.
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- Probability in the Engineering and Informational Sciences , Volume 18 , Issue 3 , July 2004 , pp. 395 - 409
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- © 2004 Cambridge University Press
1. INTRODUCTION
During the past several decades, various concepts of stochastic comparisons between random variables have been defined and studied in the literature, because they are useful in modeling for reliability and economics applications and as mathematical tools for proving important results in applied probability (see Shaked and Shanthikumar [20] for an exhaustive monograph on this topic).
For any life variable X ≥ 0, the residual life variable Xt = [X − t|X ≥ t], where t ∈ (0,lX) and lX = sup{t : FX(t) < 1}, is a nonnegative random variable representing the remaining life of X at age t. Hence, if F(·) is the distribution function of X and F(·) ≡ 1 − F(·) is its survival function, then the survival function of Xt is given by
Given two random variables X and Y, X is said to be smaller than Y in the hazard rate order (denote by X ≤HR Y) if
where the stochastic ordering (≤st) means that FXt(t) ≤ FYt(t) for all t.
However, it is reasonable to presume that in many realistic situations, the random life variable is not necessarily related to the future but can also refer to the past. For instance, consider a system whose state is observed only at certain preassigned inspection times. If at time t the system is inspected for the first time and it is found to be “down,” then the failure relies on the past (i.e., on which instant in (0,t) it has failed). It thus seems natural to study a notion that is dual to the residual life, in the sense that it refers to past time and not to future (see Di Crescenzo and Longobardi [9]).
For any random variable X, let
denote a random variable whose distribution is the same as the conditional distribution of t − X given that X < t. When the random variable X denotes the lifetime (X ≥ 0, with probability 1) of a unit, X(t) is known as the inactivity time or reversed residual life (see, for instance, Chandra and Roy [8], Block, Savits, and Singh [7], Li and Lu [14], and Nanda, Singh, Misra, and Paul [17]).
Now we recall the definition of the mean inactivity time order (≤MIT), the increasing concave order (≤ICV), and the reversed hazard rate order (≤RHR).
Definition 1.1: Let X and Y be two nonnegative random variables with absolutely continuous distribution functions F and G and densities f and g, respectively. X is said to be smaller than Y in the following:
(i) the mean inactivity time order (denoted by X ≤MIT Y) if
(ii) the increasing concave order (denoted by X ≤ICV Y) if
(iii) the reversed hazard rate order (denoted by X ≤RHR Y) if
For more details, one may refer to Shaked and Shanthikumar [20] and Muller and Stoyan [16] for the stochastic order (≤ST), hazard rate order (≤HR), reversed hazard rate order (≤RHR), increasing concave order (≤ICV) and Nanda et al. [17] for the mean inactivity time order (≤MIT) and other commonly used stochastic orders.
The purpose of this article is to study several preservation properties of stochastic comparisons based on the mean inactivity time order. Section 2 contains definitions, notations, and basic properties used throughout the article. Also in that section, we give some characterizations and relationships of the ≤MIT order and other well-known orders. In Section 3, we present some preservation results under the operations of convolution and mixture, as well as some examples of interest in reliability theory. Finally, in Section 4, we discuss hypothesis testing in the increasing mean inactivity time class (IMIT) defined as the class where E [X(t)] is increasing for all nonnegative t.
Throughout the article we will use the term “increasing” in place of “nondecreasing” and “decreasing” in place of “nonincreasing.” a/0 is understood to be ∞ whenever a > 0. All integrals and expectations are implicitly assumed to exist whenever they are written.
2. DEFINITIONS, NOTATIONS, AND CHARACTERIZATIONS
Let X and Y have the distribution functions F and G, respectively; we denote the expected value of the random variables X(t) and Y(t) by α(t) and β(t), respectively, where
Observe that by the definition of ≤MIT order, X ≤MIT Y holds if and only if α(t) ≥ β(t) for all t ≥ 0. Actually, an equivalent condition for MIT order is given in Nanda et al. [17], and is the following.
Proposition 2.1: Let X and Y be two continuous nonnegative random variables with absolutely continuous distribution functions F and G. Then, X ≤MIT Y if and only if
For any real number a, let a− denote the negative part of a; that is, a− = a if a ≤ 0 and a− = 0 if a > 0. Therefore, if X ≤MIT Y, then
or, equivalently, X ≤MIT Y if and only if
Nanda et al. [17] proved that the reversed hazard rate order is stronger than the mean inactivity time order. In the following result, we prove that the mean inactivity time order is stronger than the increasing concave order.
Theorem 2.1: Let X and Y be two nonnegative random variables. If X ≤MIT Y, then X ≤ICV Y.
Proof: Let F and G be the distribution functions of X and Y, respectively. From (2.1), it follows that
Therefore,
Now, the proof is similar to that of Theorem 3.A.13 in Shaked and Shanthikumar [20]. A straightforward computation gives
Thus,
Since X ≤MIT Y implies that
it follows that (1.1) holds. █
We have the following implications among some of the previous orders:
3. PRESERVATION RESULTS
Useful properties of the stochastic orders are their closure with respect to typical reliability operations like convolution or mixture (see Barlow and Proschan [5] and Shaked and Shanthikumar [20]). In this section, we present some preservation results for the mean inactivity time order (≤MIT). First, we recall the definition of some notions that will be used in the sequel.
Definition 3.1: Given two continuous random variables X and Y with densities f and g, respectively, X is said to be smaller than Y in the likelihood ratio order (denoted by X ≤lr Y) if
Definition 3.2: A probability vector α = (α1,…,αn) with αi > 0 for i = 1,2,…,n is said to be smaller than the probability vector β = (β1,…,βn) in the sense of the discrete likelihood ratio order, denoted by α ≤dlr β, if
Definition 3.3: A function g(x), −∞ < x < ∞, is said to be a Polya function of order 2 (PF2) if the following hold:
(a) g(x) ≥ 0 for −∞ < x < ∞ and
(b)
for all −∞ < x1 < x2 < ∞ and −∞ < y1 < y2 < ∞ or, equivalently,
(b′) log[g(x)] is concave on (−∞,∞).
The equivalence of (b) and (b′) is shown in Barlow and Proschan [5, Exercise12, p.79].
3.1. Convolution
As an important reliability operation, convolutions of a certain stochastic order are often paid much attention. The closure properties of ≤ST, ≤HR, ≤RHR, and ≤ICV orders can be found in Shaked and Shanthikumar [20]. In Theorem 3.1, we establish the closure property of the ≤MIT order under the convolution operation. In general, if X1 ≤MIT Y1 and X2 ≤MIT Y2, where X1 and X2 are independent random variables and Y1 and Y2 are also independent random variables, then it is not necessarily true that X1 + X2 ≤MIT Y1 + Y2. However, if these random variables have log-concave density, then it is true. This is shown in the following.
Theorem 3.1: Let X1, X2, and Y be three nonnegative random variables, where Y is independent of both X1 and X2, and let Y have density g. If X1 ≤MIT X2 and g is log-concave, then X1 + Y ≤MIT X2 + Y.
Proof: First, we note that for fixed s ≥ 0 and i = 1,2,
As shown in Proposition 2.1, the assertion follows if we prove that Φ(i,t) is TP2 in (i,t) (Joag-Dev, Kochar, and Proschan [12]). By the assumption X1 ≤MIT Y1, we can say that ψ(i,z) is TP2 in (i,z). Moreover, since Y has log-concave density, fY(t − z) is TP2 in (t,z). Therefore, by the basic composition formula (Karlin [13]), it follows that Φ(i,t) is TP2 in (i,t). This completes the proof. █
Corollary 3.1: If X1 ≤MIT Y1 and X2 ≤MIT Y2, where X1 is independent of X2 and Y1 is independent of Y2, then the following statements hold:
(i) If X1 and Y2 have log-concave densities, then X1 + X2 ≤MIT Y1 + Y2.
(ii) If X2 and Y1 have log-concave densities, then X1 + X2 ≤MIT Y1 + Y2.
Proof: The following chain of inequalities, which establish (i), follows from Theorem 3.1:
The proof of (ii) is similar. █
Theorem 3.2: If X1, X2, … and Y1, Y2, … are sequences of independent random variables with Xi ≤MIT Yi and Xi and Yi have log-concave densities for all i, then
Proof: We will prove the theorem by induction. Clearly, the result is true for n = 1. Assume that the result is true for p = n − 1; that is,
Note that each of the two sides of (3.1) has a log-concave density (see, e.g., Karlin [13, p.128]). Appealing to Corollary 3.1, the result follows. █
3.2. Mixture
Let now X(θ) be a random variable having distribution function Fθ and let Θi be a random variable having distribution Gi, for i = 1,2, and support R+. The following is a closure of MIT order under mixture.
Theorem 3.3: Let {X(θ), θ ∈ R+} be a family of random variables independent of Θ1 and Θ2. If Θ1 ≤lr Θ2 and if X(θ1) ≤MIT X(θ2) whenever θ1 ≤ θ2, then X(Θ1) ≤MIT X(Θ2).
Proof: Let Fi be the distribution function of X(Θi), with i = 1,2. We known that
Because of Proposition 2.1, we should prove that
is TP2 in (i,t). However, actually
By the assumption X(θ1) ≤MIT X(θ2) whenever θ1 ≤ θ2, we have that ψ(θ,t) is TP2 in (θ,t), and from the assumption Θ1 ≤lr Θ2, it follows that gi(θ) is TP2 in (i,θ). Thus, again, the assertion follows from the basic composition formula. █
Suppose that Xi, i = 1,…,n, is a collection of independent random variables. Suppose that Fi is the distribution function of Xi. Let α = (α1,…,αn) and β = (β1,…,βn) be two probability vectors. Let X and Y be two random variables having the respective distribution functions F and G defined by
The following result gives conditions under which X and Y are comparable with respect to the MIT order. One can refer to Ahmed [3] and Ahmed and Kayid [4] for a similar preservation property of the mean residual life order (≤MRL) and the Laplace transform of residual life order (≤Lt−rl), respectively. Definition, properties, and applications of ≤MRL order and ≤Lt−rl order can be found, for instance, in Shaked and Shanthikumar [20], Belzunce, Ortega, and Ruiz [6], and Gao, Belzunce, Hu, and Pellerey [10].
Theorem 3.4: Let X1,…,Xn be a collection of independent random variables with corresponding distribution functions F1,…,Fn, such that X1 ≤MIT X2 ≤MIT … ≤MIT Xn and let α = (α1,…,αn) and β = (β1,…,βn) such that α ≤dlr β. Let X and Y have distribution functions F and G defined in Eqs. (3.2). Then, X ≤MIT Y.
Proof: Again, because of Proposition 2.1, we need to establish that
Multiplying by the denominators and canceling out equal terms, it can be shown that inequality (3.3) is equivalent to
or, equivalently
Now, for each fixed pair (i,j) with i < j, we have
which is nonnegative because both terms are nonpositive by assumption. This completes the proof. █
3.3. Applications
To demonstrate the usefulness of the above results in recognizing MIT-ordered random variables, we consider the following examples.
Example 3.1: Let Xλ denote the convolution of n exponential distributions with parameters λ1,…,λn, respectively. Assume without loss of generality that λ1 ≤ ··· ≤ λn. Since exponential densities are log-concave, Theorem 3.4 implies that Xλ ≤MIT Yμ whenever λi ≥ μi for i = 1,…,n.
Example 3.2: Let Xi ∼ exp(λi), i = 1,…,n, be independent random variables. Let X and Y be α and β mixtures of Xi's. An application of Theorem 3.4 is immediately X ≤MIT Y for every two probability vectors α and β such that α ≤dlr β.
Another application of Theorem 3.4 is contained in following example.
Example 3.3: Let Xλ and Xμ be as given in Example 3.1. For 0 ≤ q ≤ p ≤ 1 and p + q = 1, we have
4. TESTING IN THE IMIT CLASS
In the literature, many nonparametric classes of distributions have been defined (cf. Barlow and Proschan [5] and Ross [18]). In particular, the decreasing reversed hazard rate (DRHR) class of distributions has been studied by many researchers in the recent past (cf. Shaked and Shanthikumar [20], Block et al. [7], and Sengupta and Nanda [19]). Recently, Chandra and Roy [2] introduced a new nonparametric class called the increasing mean inactivity time class of life distributions (abbreviated as the IMIT class).
Recall that a random variable X having distribution function F(·), density f (·), and reversed hazard rate function
is said to have the following:
1. Increasing mean inactivity time (IMIT) if E [X(t)] is increasing in t > 0
2. Decreasing reversed hazard rate (DRHR) if
is decreasing in t > 0
Block et al. [7] have shown that there exists no nondegenerate life distribution that has increasing reversed hazard rate (IRHR) over the domain [0,∞). Also, Nanda et al. [17] proved that there exists no nonnegative random variable for which E [X(t)] decreases over the domain [0,∞) and the DRHR property is stronger than the IMIT property; that is,
On the other hand, in the context of reliability, “ageless notion” is equivalent to the phenomenon that age has no effect on the residual survival function of a unit. Ageless has thus been equivalently described as constant failure rate, constant mean residual life, and exponential survival distribution. Note that the exponential distribution defines this notion; hence, testing any aging notion is done via testing exponentiality versus the class at hand. This applies to many classes, such as increasing hazard rate (IHR), increasing hazard rate average (IHRA), new better than used (NBU), new better than used in expectation (NBUE), harmonic new better than used in expectation (HNBUE), and decreasing mean residual lifetime (DMRL); see Ahmad [1], Ahmad and Mugdadi [2], and Mugdadi and Ahmad [15] for recent developments and references.
In order to do testing for the IMIT class (or the DRHR for that matter), one observes that there is no boundary distribution at all (i.e., there is no distribution where MIT (or RHR) is constant). The exponential distribution is easily seen to have IMIT and DRHR. Hence, to do testing for IMIT (or DRHR), we test H0 : F = F0 against H1 : F is IMIT (or DRHR) and not F0, where F0 is known (up to a set of parameters). One obvious choice of F0, of course, is the exponential. Thus, we address H0 : F is exponential (μ) against H1 : F is IMIT and not exponential.
Note that F ∈ IMIT if and only if
we thus take as a measure of departure from H0,
Note that if F is exp(μ), then
.
Let X1,…,Xn denote a random sample from F. A nonparametric estimate of δ(1) is
where k(·) is a known probability density function which is bounded and symmetric with mean 0 and finite variance σk2 and a = an are positive constant such that a → 0, an → ∞ as n → ∞.
The following theorem gives the large sample behavior of
both under H0 and in general.
Theorem 4.1: If na4 → 0 as n → ∞, if f has a bounded second derivative, and if V(ψn(X1)) < ∞, where ψn(X1) is given in Eq. (4.10), then
is asymptotically normal with mean 0 and variance limn→∞ V(ψn(X1)). Under H0, the variance is
.
Proof: First, it is easy to see that with gn(x) = E(1/a)k((x − X)/a),
We can also write
Thus,
However,
Hence,
Next, let us look at
, writing Θn for
and
Then, using a standard decomposition, we have
where
Now, by the Layaponoulf central limit theorem, the first term of Eq. (4.7) is asymptotically normal if
However, using the fact that gn(x) = f (x) + (a2/2) f′′(x)σk2, we easily see that for large n,
and
Hence, for n large enough,
where
Hence, V(ψα(X1)) = V(n(X1)) + 0(a4) and E|ψ(X1)|3 ≤ C3. Therefore, Ln → 0, provided that na4 → 0.
Finally,
The null variance is obtained by substituting the exponential into η(X1). The result now follows. █
To conduct the test, calculate
and reject H0 if this is much larger than Zα. Of course, we must choose k and a to carry out the test. The choice of k is not crucial and the standard normal will do fine. The choice of a is crucial and there are different ways to do that (cf. Wand and Jones [21]). The easiest and highly practical rule is the normal scale rule (Wand and Jones [21, p.60]) with a = cn−1/α, with α an integer greater than 1 and c is the sample standard deviation.
To assess the goodness of this test, one can evaluate its limiting Pitman efficacy for an alternative that is IMIT but not exponential and compare it to similar values of other tests for this problem. Because there are no other tests known for this problem, this comparison is left to future work on the topic. The asymptotic Pitman efficacy of a test based on a measure of departure from H1 equal to δ(1) is given by
Two of the distributions that are IMIT but are not exponential are as follows:
1. The linear failure rate:
1. The Makeham:
Note that the exponential is attained at θ = 0 in both cases. The APE of the above test is
where
Direct calculation gives a value of APE equal to 0.732 and 0.244 for the above two alternatives, respectively. These efficacy values are to be compared to those of any other procedure that might be developed for this problem. These values, however, are very close to the values obtained in standard life testing problems such as testing for decreasing mean residual lifetime (DMRL), where the Hollander and Proschan [11] test has values 0.866 and 0.242, respectively. Note, however, that these two problems are not compatible.
The interested reader might want to choose other distributions as null distributions and develop the null value of δ(1) and perform the test after the null variance is calculated.
Acknowledgment
The authors thank Professor Franco Pellerey for his useful comments and suggestions regarding the first draft of this work, which helped improve the presentation and content.
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