Asok K. Nanda

Log-concave and concave distributions in reliability

Nonparametric classes of life distributions are usually based on the pattern of aging in some sense. The common parametric families of life distributions also feature monotone aging. In this paper we consider the class of log-concave distributions and the subclass of concave distributions. The work is motivated by the fact that most of the common parametric models of life distributions (including Weibull, Gamma, log-normal, Pareto, and Gompertz distributions) are log-concave, while the remaining life of maintained and old units tend to have a concave distribution. The classes of concave and log-concave distributions do not feature monotone aging. Nevertheless, these two classes are shown to have several interesting and useful properties. We examine the closure of these classes under a number of reliability operations, and provide sharp reliability bounds for nonmaintained and maintained units having life distribution belonging to these classes.

Some results on generalized residual entropy

Shannons entropy plays an important role in the context of information theory. Since this entropy is not applicable to a system which has survived for some units of time, the concept of residual entropy has been developed in the literature. Here we generalize the residual entropy by choosing a convex function @f with @f(1)=0. In this paper, some orderings and aging properties have been defined in terms of the generalized residual entropy function and their properties have been studied. Quite a few results available in the literature have been generalized and some distributions (viz. uniform, exponential, Pareto, power series, finite range) have been characterized through the generalized residual entropy.

The Hazard Rate and the Reversed Hazard Rate Orders, with Applications to Order Statistics

In this paper we first point out a simple observation that can be used successfully in order to translate results about the hazard rate order into results about the reversed hazard rate order. Using it, we derive some interesting new results which compare order statistics in the hazard and in the reversed hazard rate orders; as well as in the usual stochastic order. We also simplify proofs of some known results involving the reversed hazard rate order. Finally, a few further applications of the observation are given.

SOME RESULTS ON REVERSED HAZARD RATE ORDERING

Recently, the reversed hazard rate (RHR) function, defined as the ratio of the density to the distribution function, has become a topic of interest having applications in actuarial sciences, forensic studies and similar other fields. Here we establish results with respect to RHR ordering between the exponentiated random variables. We also address the ordering results between component redundancy and system redundancy. Both the cases of matching spares and non-matching spares are discussed. In case of matching spares, a sufficient condition has been given for component redundancy to be superior to the system redundancy with respect to the reversed hazard rate ordering for any coherent system.

Preservation of some likelihood ratio stochastic orders by order statistics

We show that the order statistics, in a sample from a distribution that has a logconcave density function, are ordered in the up shifted likelihood ratio order. We also show that the order statistics from two different collections of random variables are ordered in the up shifted likelihood ratio order or in the regular likelihood ratio order, if the underlying random variables are so ordered. Some results about the down shifted likelihood ratio order are also included in this paper. Finally it is indicated how the results can be applied in reliability theory.

Some weighted distribution results on univariate and bivariate cases

In this paper, some partial ordering results regarding the original and the weighted distributions of random variables and random vectors have been derived. Bivariate weighted distributions have been discussed and some results have been obtained regarding them. Some dependence properties have also been studied.

Some Reliability Properties of the Inactivity Time

In this article, it is shown that, apart from the expected inactivity time, the second-order moment of inactivity time determines the distribution uniquely. We also show that any one partial moment, fractional or integral, of inactivity time uniquely determines a distribution. Different properties of the inactivity time of the components of a parallel system, at the system level, have been studied. Stochastic comparisons between two parallel systems based on this is also studied here.

Some Shifted Stochastic Orders

In this paper we study some known and some new shifted orders. We compare them, we obtain some basic properties of them, we derive some closure properties of them, and we show how they can be used for stochastic comparisons of order statistics; that is, of k-out-of-n systems.

On Generalizing Process Capability Indices

Abstract A generalized process capability index, defined as the ratio of proportion of specification conformance (or, process yield) to proportion of desired (or, natural) conformance, has been developed. Almost all the process capabilities defined in the literature are directly or indirectly associated with this generalized index. Normal as well as non-normal and continuous as well as discrete random variables could be covered by this new index. It can also be assessed under either unilateral or bilateral specifications. We deal with the proposed index in case of normal, exponential and Poisson processes. Under each distributional assumption, point estimators for the proposed index are suggested and compared through simulation study. Two real-world applications have been discussed using the proposed index.

Preservation of some partial orderings under the formation of coherent systems

The reversed (backward) hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their reversed hazard rate functions. In this paper, we have given some sufficient conditions under which the ordering between the components with respect to the reversed hazard rate is preserved under the formation of coherent systems. We have also shown that these sufficient conditions are satisfied by k-out-of-n systems. Both the cases when components are identically distributed and not necessarily identically distributed are discussed. Some results for likelihood ratio order are also obtained. The parallel (series) systems of not necessarily iid components have been characterized by means of a relationship between the reversed hazard rate (hazard rate) function of the system and the reversed hazard rate (hazard rate) functions of the components.