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We study the behavior of the failure rate associated with the distribution of a random variable of the form X = Y + U , where Y, U are independent and U has bounded support. First, we obtain monotonicity results and bounds for the failure rate of X in the case where U has a uniform distribution and, in particular we show that, asymptotically, the failure rates of X and Y tend to the same limit. Some of the results are generalized for the case where the distribution of U is not uniform, but has bounded support. Further, we show that if the failure rate of a non negative variable X is constant in some interval (L , ∞) , then X can be written as the sum of two independent random variables, one of which is exponential and the other (which is not necessarily uniform) has support [ 0 , L ] . [ABSTRACT FROM AUTHOR]

We obtain new bounds for the renewal function, as well as for the expected number of renewals over an interval (t , t + h ] . Improved bounds are given when the interarrival distribution belongs to certain aging classes. Our results are compared with existing ones in the literature, both theoretically and with the aid of numerical resuts. [ABSTRACT FROM AUTHOR]

Generalized order statistics unify the study of order statistics, record values, k-records, Pfeifer’s records and several other cases of ordered random variables. In this paper, we first provide several comparison results for generalized order statistics in terms of the dynamic cumulative residual quantile entropy order. We also prove that when the minimum of random vectors of generalized order statistics is increasing mean residual life (IMRL), the spacings of generalized order statistics are ordered by variance, and the covariance between successive spacings is nonnegative. The normalized spacings of generalized order statistics from two sample sequences are also compared with respect to the mean residual life and the excess wealth orders.

The classical multiplicative frailty model in survival analysis accounts for unobserved heterogeneity between individuals. It is of great importance to identify how the variation of the frailty variable affects that of the overall population. This paper is mainly to present how the dispersive and the excess wealth orders between two frailty variables, translate into the corresponding orders between the resulting overall population variables. For the mean residual life and the mean inactivity time orders, we also obtain relevant analogous results in multiplicative frailty models. [ABSTRACT FROM AUTHOR]

Abstract: This article investigates some properties of the mean residual life function of (n−k+1)-out-of-n systems, when the lifetimes of the system components are independent random variables but not necessarily identically distributed and when the joint distribution of the component lifetimes is exchangeable, extending the results of [On the mean residual life function of coherent systems. IEEE Transactions on Reliability 57 (4) 574–580] for the case of independent and identically distributed components. The extension to a coherent system with exchangeable components is also given. [Copyright &y& Elsevier]

Kamps (1995a,b) introduced the concept of generalized order statistics (GOSs) as a unified approach to a variety of models of random variables (rv's) arranged in ascending order of magnitude. In this note, we will examine the preservation properties of class of life distributions DRHR based on GOSs. Also, we give the transition properties of IUPL class. [ABSTRACT FROM AUTHOR]

Abstract: Generalized order statistics introduced by Kamps [1995. A concept of generalized order statistics. J. Statist. Plann. Inference 48, 1–23] provides a unified approach to a variety of models concerning ordered random variables. This paper carries out stochastic comparisons of conditional generalized order statistics in terms of the likelihood ratio order and the hazard rate order from two samples, and establishes some stochastic monotonicity properties. The main results strengthen and generalize the corresponding results established recently in the literature. Finally, some applications of the main results are given as well. [Copyright &y& Elsevier]

In many practical situations, order statistics arise naturally with random sample size. In this article, we review results on partial orderings and aging properties of such order statistics. The comparison of order statistics for different sample sizes is also discussed here. [ABSTRACT FROM AUTHOR]

Abstract: In a progressively type II censored life-testing experiment intact units may be removed from the experiment after every failure. If the initial number of units in the experiment and the total number of failures are fixed, the experimenter may choose between different censoring schemes. By specifying an optimality criterion, one may improve the outcome of the experiment by choosing the respective optimal scheme. We establish a simple property of a general optimality criterion that yields optimality of certain extremal schemes. Applications to some criteria that measure the total time of the experiment and its variability illustrate the approach. The results are based on stochastic orderings of generalized order statistics. [Copyright &y& Elsevier]

This paper investigates the stochastic comparison of spacings of record values. The likelihood ratio order, the hazard rate order and the mean residual life order between spacings of record values from one sample sequence are developed. Furthermore, the increasing convex order and the mean residual life order between simple spacings of record values from two sample sequences are built as well. [ABSTRACT FROM AUTHOR]

By assuming that a random variable X possesses an aging property, we provide conditions under which the corresponding weighted version X1, with weight function w1(·), would also possess this aging property. Similarly, by assuming that two random variables X and Y are ordered with respect to a stochastic order, we provide conditions under which the corresponding weighted versions X1 and Y2, with weight functions w1(·) and w2(·), respectively, preserve this stochastic ordering. We also point out fallacies in the similar results claimed by Jain et al. (1989) and Bartoszewicz and Skolimowska (2006) and correct them. [ABSTRACT FROM AUTHOR]

We present sufficient conditions for log-concavity and log-convexity of the residual life and the inactivity time. We also make stochastic comparisons on the residual life and the inactivity time in terms of the failure rate order, the mean residual life order and the usual stochastic order. These results strengthen some conclusions in Li and Zuo[11] and Yue and Cao[19]. [ABSTRACT FROM AUTHOR]

Abstract: Let denote the order statistics from a sample of n independent random variables , all identically distributed as some with support . For a given positive integer m, , let , , denote the m-spacings of the X-sample, here . It is shown that if X has a logconvex [logconcave] density, then for , is smaller [larger] than in the likelihood ratio order. It is also shown that if X has a logconcave density then, for , is smaller than in the likelihood ratio order. If, instead, X is assumed to have an increasing failure rate and a decreasing reversed hazard rate, then this result can be weakened from the likelihood ratio order to the hazard rate order. We thus strengthen and complement some results in Misra and van der Meulen (J. Statist. Plann. Inference 115 (2003) 683.) and Hu and Wei (Statist. Probab. Lett. 53 (2001) 91.). [Copyright &y& Elsevier]

Let X be a nonnegative and continuous random variable having the probability density function (pdf) f(.). Let Xk:n (k=1,2,…,n) denote the kth order statistic based on n independent observations on X and, for a given positive integer m (⩽n), let Dk,n(m)=Xk+m−1:n−Xk−1:n, k=1,2,…,n−m+1, denote the successive (overlapping) spacings of gap size m (to be referred as m-spacings); here X0:n≡0. It is shown that if f(.) is log convex, then the pdf of corresponding simple (gap size one) spacings Dk,n(1), k=1,2,…,n, are also log convex. It is also shown that the m-spacings Dk,n(m), k=1,2,…,n−m+1, preserve the log concavity of the parent pdf f(.). Under the log convexity of the parent pdf f(.), we further show that, for k=1,2,…,n−m, Dk,n(m) is smaller than Dk+1,n(m) in the likelihood ratio ordering and that, for a fixed 1⩽k⩽n−m+1 and n⩾k+m−1, Dk,n+1(m) is smaller than Dk,n(m) in the likelihood ratio ordering. Finally, we show that if X has a decreasing failure rate then, for k=1,2,…,n−m, Dk,n(m) is smaller than Dk+1,n(m) in the failure rate ordering and that, for a fixed 1⩽k⩽n−m+1 and n⩾k+m−1, Dk,n+1(m) is smaller than Dk,n(m) in the failure rate ordering. [Copyright &y& Elsevier]

The paper gives existence conditions for the generalized Euler family of distributions and relates these to distributions already in the literature. The logconvexity, logconcavity, failure rate, infinite divisibility, strong unimodality, and over and under-dispersion properties are examined using the methods of Gupta et al. (J. Statist Plann. Inference 65 (1997) 225). [Copyright &y& Elsevier]

The paper gives a probabilistic proof of the Bailey–Daum theorem. It also provides random walks for certain special cases of the Bailey–Daum distribution and discusses the distribution''s moment properties, logconvexity, logconcavity, and infinite divisibility. [Copyright &y& Elsevier]

Let Y k,n denote the nth (upper) k-record value of an infinite sequence of independent, identically distributed random variables with common continuous distribution function F. We show that if the nth k-record value Y k,n has an increasing failure rate (IFR), then Y l,n ( l k+1, n+1( n≤ k+1) also have IFR distributions. On the other hand, if Y k,n has a decreasing failure rate (DFR), then Y l,n (1> k) has also a DFR distribution. We also present some results concerning log convexity and log concavity of Y k,n . [ABSTRACT FROM AUTHOR]