Abstract
In this paper, we discuss the variability ordering of lifetimes of series systems with two independent heterogeneous Weibull components in terms of the right spread order. We give some sufficient conditions implying the right spread order between lifetimes of series systems.
MSC:60E15, 62G30.
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1 Introduction
Order statistics play an important role in statistical inference, life testings, reliability theory and many other areas. Let be n random variables and let denotes their i th order statistic, . In reliability theory, the k th order statistic corresponds to the lifetime of a -out-of-n system. Parallel and series systems are the building blocks of more complex coherent systems, wherein the lifetime of a series system corresponds to the smallest order statistic and the lifetime of a parallel system corresponds to the largest order statistic . Misra and Misra [1] have considered the stochastic comparisons of series systems according to the reversed hazard rate order and the likelihood ratio order. Misra et al. [2] have compared lifetimes of series (parallel) systems arising out of different allocations of one or two standby redundancies in light of the hazard rate order, the increasing concave (convex) order, and the stochastic precedence order. Moreover, other authors have studied stochastic comparisons of lifetimes of series and parallel systems; for example, see [3–12], and references cited therein.
To continue our discussion, we need definitions of some stochastic orders and the concept of majorization which is given in Section 2. Zhao and Balakrishnan [4] have focused on comparison the largest order statistic in terms of the important right spread order from two heterogeneous exponential variables, that is, the lifetime of a parallel system with two independent heterogeneous exponential components. Let , be independent exponential random variables with having hazard rate , . Let , be another set of independent exponential random variables with having hazard rate , . Zhao and Balakrishnan [4] showed that under the condition
Weibull distribution is commonly used lifetime distribution in modeling lifetime data. Let X be a Weibull random variable with shape parameter α and scale parameter λ, and with probability density function
Recently, Prabhakar Murthy et al. [13] have studied Weibull models in detail. The exponential distribution and the Rayleigh distribution can be obtained from Weibull distribution by putting and , respectively.
Let be a vector of independent Weibull random variables with common shape parameter and respective scale parameters , . Let be another vector of independent Weibull random variables with common shape parameter α and scale parameter λ. When , Khaledi and Kochar [6] proved that for . Let denote the minimum of , . When , Fang and Zhang [12] proved that for . Fang and Zhang [11] showed that ; and if , then ; and if , then . Let be independent random variables with common shape parameter and respective scale parameters , . Fang and Zhang [11] showed that implies that for and for .
In this paper, we compare the lifetimes of series systems with respect to the right spread order arising from two independent heterogeneous Weibull random variables. Let be a vector of independent Weibull random variables with common shape parameter and respective scale parameters and , and let be another vector of independent Weibull random variables with common shape parameter α and respective scale parameters and . Under the condition , we will show that if then ; and if then . On the other hand, under the condition
or
we establish .
2 Preliminaries
Suppose that the random variables X and Y have distribution functions and , survival functions and , the right continuous inverse functions and , density functions and , and the reversed hazard rate functions and , respectively. In this paper, all the random variables are assumed to be nonnegative and having support . Let .
Throughout this paper, the notions increasing and decreasing mean nondecreasing and nonincreasing, respectively.
Definition 2.1 X is said to be
-
(i)
less dispersive than Y if
for , in symbols, ;
-
(ii)
less right spread than Y if
for , in symbols, ;
-
(iii)
smaller than Y in the reversed hazard rate order if for all , or equivalently, if is decreasing in ; in symbols, ;
-
(iv)
smaller than Y in the convex transform order if is convex in x on the support of X, in symbols, .
The dispersive order and the right spread order are two basic partial orders comparing the variabilities in the two distributions. The right spread order has been widely used in many fields, especially in economics, insurance, and reliability theory. It is well known that
Several notions of stochastic orders of varying degree of strength have been discussed in detail in [14].
Majorization is a very interesting topic in statistics, which is a pre-ordering on vectors by sorting all components in increasing order.
Definition 2.2 Let , denote two n-dimensional real vectors. Let , be their ordered components. is said to be majorized by λ, in symbols , if ; for , and .
For more details on majorization and their applications, one may refer to Marshall and Olkin [15].
The following two useful results, which established an equivalent characterization of dispersive order and the right spread order in one parameter family, will be needed to prove our main results.
Lemma 2.3 (Saunders and Moran [16])
Let be a class of distribution functions, such that is supported on some interval and has a density which does not vanish on any subinterval of , where and denote the left and right end points, respectively. Then , , , if and only if is decreasing in x, where is the derivative of with respect to a.
Lemma 2.4 Let be assumed as Lemma 2.3. Then
-
(i)
, , , if and only if is increasing in x;
-
(ii)
, , , if and only if is decreasing in x;
where is the derivative of with respect to a.
The proof of this lemma is similar to the proof of Lemma 3.1 of Kochar and Xu [17], so we omit it here.
3 Main results
Firstly, we consider stochastic comparison on the lifetimes of series systems arising from two independent heterogeneous Weibull random variables in terms of dispersive order.
Theorem 3.1 Let be a vector of independent Weibull random variables with common shape parameter and respective scale parameters λ and , and let be another vector of independent Weibull random variables with common shape parameter α and respective scale parameters λ and . Suppose that . If , then .
Proof Suppose . Denote and , we have . The distribution function of , denoted by , is given by
Taking the derivative with respect to a, we have
Taking the derivative with respect to x, we obtain the density function of as
From (3.1) and (3.2), we see that
is decreasing in . From Lemma 2.3, we can conclude that . □
Next, we provide some sufficient conditions on the lifetimes of series systems arising from two independent heterogeneous Weibull random variables in terms of the right spread order.
Theorem 3.2 Let be a vector of independent Weibull random variables with common shape parameter and respective scale parameters and , and let be another vector of independent Weibull random variables with common shape parameter α and respective scale parameters and . Suppose that .
-
(i)
If , then ;
-
(ii)
if , then .
Proof Without loss of generality, let us assume that and . Since , we have and . Let , , . We then have , , .
The survival function of , denoted by , can be expressed as
Let
Taking the derivative with respect to λ, we have
So, using (3.3) and (3.4), we obtain
where
By taking the derivative with respect to x, we have
Thus, is increasing in .
Since , then , which implies is decreasing in . On the other hand, if , then , which implies is increasing in . Now the proof follows from Lemma 2.4. □
Theorem 3.3 Let be a vector of independent Weibull random variables with common shape parameter and respective scale parameters and , and let be another vector of independent Weibull random variables with common shape parameter α and respective scale parameters and . Suppose that and , then .
Proof Without loss of generality, we assume that and . Let , , . We then have , , and .
, the survival function of , can be expressed as
So, we obtain
Taking the derivative with respect to λ, we have
By (3.5) and (3.6), we obtain
where
By taking the derivative with respect to x, we have
Thus, is increasing in .
Since and , it is easy to see that , hence , implying . Therefore, is increasing in . So, the theorem follows from Lemma 2.4(i). □
Theorem 3.4 Let be a vector of independent Weibull random variables with common shape parameter and respective scale parameters and , and let be another vector of independent Weibull random variables with common shape parameter α and respective scale parameters and . Suppose that
-
(i)
and ; or
-
(ii)
.
Then .
Proof Without loss of generality, let us assume that and . Then Assumptions (i) and (ii) provide
and
respectively.
So, the results for the case when can be obtained immediately from Theorem 3.3. In what follows, we consider that . Let , then
Let be a vector of independent Weibull random variables with common shape parameter α and respective scale parameters and . From Theorem 3.3, it follows that . In addition, we have using Theorem 3.1 which further implies . Thus we have . □
Finally, we present some applications of the results established above. In reliability analysis, we all know the lifetime of a k-out-of-n system can be represented as . Reliability engineers are not only interested in controlling the mean lifetime of a system, but also comparing the variability of the lifetimes of two systems. The lifetime of a series system corresponds to the smallest order statistic . Misra and Misra [1] have compared the lifetimes of series systems according to the reversed hazard rate order and the likelihood ratio order. We have considered the lifetimes of series systems with two independent heterogeneous Weibull components in terms of the right spread order. The results lead to the following insight with regard to n-out-of-n systems.
Example 3.5 One n-out-of-n system A with independent heterogeneous Weibull components , which have common shape parameter and scale parameter vector , the other n-out-of-n system B with independent heterogeneous Weibull components , which have common shape parameter and scale parameter vector .
-
(a)
Suppose that , . If , then the system A has a smaller variability than B by Theorem 3.2(i); if , then the system B has a smaller variability than A by Theorem 3.2(ii).
-
(b)
Suppose that , . Then the system B has a smaller variability than A by Theorem 3.3.
-
(c)
Suppose that , . Then the system B has a smaller variability than A by Theorem 3.4(i).
-
(d)
Suppose that , . Then the system B has a smaller variability than A by Theorem 3.4(ii).
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Acknowledgements
This research is supported by the National Natural Science Foundation of China (No. 11071045, 11201003); the National Statistical Science Research Project of China (No. 2012LY158); the Provincial Natural Science Research Project of Anhui Colleges (No. KJ2013A137); the National Natural Science Foundation of Anhui Province (No. 1408085MA07).
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The main idea and writing of this paper were carried out by LF. WT performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Fang, L., Tang, W. On the right spread ordering of series systems with two heterogeneous Weibull components. J Inequal Appl 2014, 190 (2014). https://doi.org/10.1186/1029-242X-2014-190
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DOI: https://doi.org/10.1186/1029-242X-2014-190