Reliability Properties of Systems with Two Exchangeable Log-Logistic Components

In this article, we study the reliability properties of systems under bivariate log-logistic model which comes out from a particular stress-strength analysis. For this model, we obtain basic reliability characteristics of series and parallel systems and investigate their properties. We also derive distribution and moments of cold standby system under the abovementioned exchangeable model.     

On the Percentile Residual Lifetime of Parallel Systems

In this paper, we consider a parallel system consisting of n components. Then, the percentile residual lifetime of the system given survival of at least n ? r + 1, r = 1, 2, …, n component(s) has been introduced, and some properties of this measure have been investigated. We show that the system accommodates decreasing percentile residual lifetime function, provided the components have increasing hazard rate functions. Different parallel systems have been compared with each other in terms of the introduced measure. Furthermore, behavior of the percentile residual lifetime of the system and the components have been compared in terms of some reliability notions. Also, a characterization result has been presented.     

On the quantile past lifetime of the components of the parallel systems

ABSTRACT

Let T_{1: n} ? T_{2: n} ? ??? ? T_{n: n} be ordered lifetimes of components of a parallel system. In this article, the α-quantile past lifetime from the failure of the component with lifetime T_{r: n} provided that the system has failed at or before time t has been introduced. Then, some properties of this measure have been studied.     

A Note on the Regression of Order Statistics

This paper examines the relationships between the mean residual life functions of parallel and k-out-of-n systems with the regression of order statistics. Using these relationships, the results and properties about the mean residual life function of those systems can be used for the regression of order statistics and vice versa. Finally, the paper proposes a definition for the mean residual life function of a k-out-of-n system when the number of failed components of the system is known.     

MRL ordering of parallel systems with two heterogeneous components

In this paper, we investigate ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components in terms of the mean residual life order. We establish, among others, that the reciprocal majorization order between parameter vectors implies the mean residual life order between the lifetimes of two parallel systems. We then extend this result to the proportional hazard rate models.     

Improved Ordering Results for Fail-Safe Systems with Exponential Components

Let X_{2: n} and Y_{2: m} be the second order statistics from n independent exponential variables with hazards λ₁, …, λ_n, and an independent exponential sample of size m with hazard change to λ, respectively. When m ? n, we obtain necessary and sufficient conditions for comparing X_{2: n} and Y_{2: m} in mean residual life, dispersive, hazard rate, and likelihood ratio orderings based on some inequalities between λ_i’s and λ. The established results show how one can compare an (n ? 1)-out-of-n system consisting of heterogeneous components with exponential lifetimes with any (m ? 1)-out-of-m system consisting of homogeneous components with exponential lifetimes.     

Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

On a Characterization of Generalized Pareto Distribution Based on Generalized Order Statistics

In recent years, several attempts have been made to characterize the generalized Pareto distribution (GPD) based on the properties of order statistics and record values. In the present article, we give a characterization result on GPD based on the spacing of generalized order statistics.     

Nonparametric estimation of mean residual functions

Situations frequently arise in practice in which mean residual life (mrl) functions must be ordered. For example, in a clinical trial of three experiments, let e (1), e (2) and e (3) be the mrl functions, respectively, for the disease groups under the standard and experimental treatments, and for the disease-free group. The well-documented mrl functions e (1) and e (3) can be used to generate a better estimate for e (2) under the mrl restriction e (1) < or = e (2) < or = e (3). In this paper we propose nonparametric estimators of the mean residual life function where both upper and lower bounds are given. Small and large sample properties of the estimators are explored. Simulation study shows that the proposed estimators have uniformly smaller mean squared error compared to the unrestricted empirical mrl functions. The proposed estimators are illustrated using a real data set from a cancer clinical trial study.     

Some characterizations of multivariate distributions using products of the hazard gradient and mean residual life components

We give a general procedure to characterize multivariate distributions by using products of the hazard gradient and mean residual life components. This procedure is applied to characterize multivariate distributions as Gumbel exponential, Lomax, Burr, Pareto and generalized Pareto multivariate distributions. Our results extend the results of several authors and can be used to study how to extend univariate models to the multivariate set-up.     

On some generalized orderings: In the spirit of relative ageing

ABSTRACT

We introduce some new generalized stochastic orderings (in the spirit of relative ageing) which compare probability distributions with the exponential distribution. These orderings are useful to understand the phenomenon of positive ageing classes and also helpful to guide the practitioners when there are crossing hazard rates and/or crossing mean residual lives. We study some characterizations of these orderings. Inter-relations among these orderings have also been discussed.     

On the mean residual life of a generalized k-out-of-n system

A generalized k-out-of-n system consists of N modules in which the i th module is composed of n_i components in parallel. The system failswhen at least f components in the whole system or at least k consecutive modules have failed. In this article, we obtain the mean residual life function of such a generalized k-out-of-n system under different conditions, namely, when the number of components in each module is equal or unequal and when the components of the system are independent or exchangeable.     

A new measure of association for bivariate survival data

Time dependent association measures between variables are of interest in bivariate survival data. Several such measures have been proposed in literature for the modelling and analysis of survival data. In this paper, we introduce a new measure of association for bivariate survival data using product moment residual life function and mean residual life function. Various properties of the proposed measure and its relationship with existing measures are discussed. We also develop a non-parametric estimator of the measure and study its asymptotic properties. The application of the result is illustrated using a real life data. Finally, a stimulation study is carried out to assess the performance of the estimator.     

On the determination of mixing density in reliability studies

In reliability studies the three quantities (1) the survival function, (2) the failure rate and (3) the mean residual life function are all equivalent in the sense that given one of them, the other two can be determined. In this paper we have considered the class of exponential type distributions and studied its mixture. Given any one of the above mentioned three quantities of the mixture a method is developed for determining the mixing density. Some examples are provided as illustrations. Some well known results follow trivially.     

Reliability functions of bivariate distributions with second kind of beta conditionals

The purpose of this article is to study the bivariate distributions with second kind of beta conditionals, presented by Castillo and Sarabia, from a reliability point of view. The reliability functions, viz the survival function, the failure rate and mean residual life function, are derived and their monotonicity is discussed for the univariate model as well as the conditional model. Finally, Clayton’s association measure is obtained in terms of the hazard gradient and its value, in the case of our model, is derived.     

Characterization of the Pearson system of distributions based on reliability measures

LetX be a random variable andX ^(w) be a weighted random variable corresponding toX. In this paper, we intend to characterize the Pearson system of distributions by a relationship between reliability measures ofX andX ^(w), for some weight functionw>0.     

On study of dynamic survival and cumulative past entropies

Abstract

Recently, a new class of measure of uncertainty, called “dynamic survival entropy”, has been defined and studied in the literature. Based on this entropy, DSE(α) ordering, IDSE(α), and DDSE(α) classes of life distributions are defined and some results are studied. In this paper, our main aim is to prove some more results of the ordering and the aging classes of life distributions mentioned above. Some important distributions such as exponential, Pareto, Pareto II, and finite range distributions are also characterized. Here we have defined cumulative past entropy and proved some interesting results.     

Attainability of the Upper Bounds for the Mean Past Lifetime of Parallel System Components

Among reliability systems, one of the basic systems is a parallel system. In this article, we consider a parallel system consisting of n identical components with independent lifetimes having a common distribution function F. Under the condition that the system has failed by time t, with t being 100pth percentile of F(t = F ^?1(p), 0 < p < 1), we characterize the probability distributions based on the mean past lifetime of the components of the system. These distributions are described in the form of a specific shape on the left of t and arbitrary continuous function on the right tail.     

Properties of systems with two exchangeable Pareto components

The life lengths of the units in a system can be modelled by a bivariate distribution. In this paper, we suppose that the joint distribution of the units is a symmetric bivariate Pareto (Lomax) distribution. For this model, we obtain basic reliability properties for series and parallel systems. J. M. Ruiz Partially Supported by Ministerio de Ciencia y Tecnologia under grant BFM2003-02947 and Fundacion Seneca under grant 00698/PI/04.     

On Redundancy Allocation to Series and Parallel Systems of Two Components

Abstract

For two components and one standby redundancy, we develop a characterization on the hazard rate order and the reversed hazard rate order of the redundant system lifetime in the context of mutually independent components lifetimes. Also, the likelihood ratio order is derived on the lifetime of the series system with two components lifetimes and two matched active redundancies lifetimes both following the proportional hazard model.