Tests for properties of residual life

The lOOα -percentile (0 < α < 1) residual life function at time x is defined to be the lOOα -percentile of the remaining life given survival up to time x. Joe and Proschan (1982b) develop tests for testing the alternatives representing decreasing 100α-percentile residual life (DPRL-α ) and the property ‘new better than used with respect to the lOOα -percentile’ (NBUP-α ). In this paper, tests are developed for DPRL[α, l) and NBUP[α, l) alternatives, where DPRL[α, l) is the class of life distributions which are DPRL-β distributions for all a ≤ β < 1 if 0 ≤ α < 1 and for all 0 < β < 1 if α = 0, and NBUP[α, l) is similarly defined. When α = 0, the DPRL[α, l) class of life distributions corresponds to the increasing failure rate class and the NBUP[α, l) class of life distributions corresponds to the new better than used class, and the test statistics are respectively asymptotically equivalent to the Hollander and Proschan (1975) test statistics for decreasing mean residual life and new better than used in expectation alternatives.     

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.     

The percentile residual life up to time t0: Ordering and aging properties

Motivated by practical issues, a new stochastic order for random variables is introduced by comparing all their percentile residual life functions until a certain instant. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are studied and also an application in reliability theory is described. Finally, we present some characterization results of the decreasing percentile residual life up to time t₀ aging notion.     

Preservation of generalized ageing classes on the residual life at random time

In this paper, we provide some new preservation properties of generalized ageing classes (s-IFR) on the residual life at random time, where s is a nonnegative integer. We also obtain bounds of the residual life at exponential random time. Results are expected to be useful in the reliability, queue theory and actuarial science.     

Some results on generalized ageing classes

In this paper we provide new results about generalized ageing classes on the excess lifetime of a renewal process. We also obtain some characterizations of generalized ageing classes by means of the residual life at random time.     

Two notions of positive ageing based on stochastic dominance

The negative effects of age on the life length of a device or positive ageing are commonly used criteria for classifying life distributions. In this paper two notions of positive ageing are considered. These are the new better (worse) than used in average conditional survival probability and harmonic new better (worse) than used in upper tail. Closure of these notions under mixture and convolution are studied. The survivals of a device subject to discrete shocks of these notions of ageing which occur according to homogeneous Poisson processes are studied. A cumulative damage model is considered. Two test statistics are proposed for the two notions of ageing.     

New multivariate aging notions based on the corrected orthant and the standard construction

ABSTRACT

Recently, some well-known univariate aging classes of lifetime distributions have been characterized by means of properties of their quantile functions and excess-wealth functions. The generalization of the univariate aging notions to the multivariate case involve, among other factors, appropriate definitions of multivariate quantiles or regression representation and related notions, which are able to correctly describe the intrinsic characteristic of the concepts of aging that should be generalized. The multivariate versions of these notions, which are characterized by using the multivariate u-quantiles and the multivariate excess-wealth function, are considered in this paper. Relationships between such multivariate aging classes are studied, and examples are provided.     

On additive-multiplicative hazards model

In survival analysis and reliability theory, a fundamental problem is the study of lifetime properties of a live organism or system. In this regard, there have been considered and studied several models based on different concepts of ageing such as hazard rate and mean residual life. In this paper, we consider an additive-multiplicative hazard model (AMHM) and study some reliability and ageing properties of the proposed model. We then specify the bivariate models whose conditionals satisfy AMHM. Several properties of the proposed bivariate model are investigated and adequacy of the model is evaluated based on a real data set.     

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.     

Tests of non-monotonic stochastic aging notions in reliability theory

Testing of various classes of life distributions has been a subject of investigation for more than four decades. In this study we restrict ourselves to the problem of testing exponentiality against non-monotonic aging notions. We model non-monotonic aging using the notions of bathtub failure rate, increasing and then decreasing mean residual life and new worse then better than used in expectation classes. The different tests of exponentiality against these alternatives are discussed in detail.     

Renyi's entropy for residual lifetime distribution

In the present paper, we introduce and study Renyi's information measure (entropy) for residual lifetime distributions. It is shown that the proposed measure uniquely determines the distribution. We present characterizations for some lifetime models. Further, we define two new classes of life distributions based on this measure. Various properties of these classes are also given.     

The role of weighted distributions in stochastic modeling

C. R. Rao pointed out that “The role of statistical methodology is to extract the relevant information from a given sample to answer specific questions about the parent population” and raised the question “What population does a sample represent”? Wrong specification can lead to invalid inference giving rise to a third kind of error. Rao introduced the concept of weighted distributions as a method of adjustment applicable to many situations.

In this paper, we study the relationship between the weighted distributions and the parent distributions in the context of reliability and life testing. These relationships depend on the nature of the weight function and give rise to interesting connections between the different ageing criteria of the two distributions. As special cases, the length biased distribution, the equilibrium distribution of the backward and forward recurrence times and the residual life distribution, which frequently arise in practice, are studied and their relationships with the original distribution are examined. Their survival functions, failure rates and mean residual life functions are compared and some characterization results are established.     

Comparing and testing order relations between percentile residual life functions

We consider the problem of comparing and testing order relations between percentile residual life functions. We introduce R-R plots and processes and develop a general approach for this problem.     

Stochastic orders and aging notions based upon the moment generating function order: Theory

The purpose of this paper is to study new notions of stochastic comparisons and aging notions based on the moment generating function order. Two characteristics of the moment generating function order of residual lives are obtained, and some closure properties of the aging classes DRL_mg (IRL_mg), NBU_mg (NWU_mg) and EBU_mg (EWU_mg) under some transformations are developed. As application, some results for random sums with applications to shock models are obtained.     

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 NBU ageing notion within the competing risks model

The classical competing risks model deals with failure times of items subject to multiple causes of failure. We propose a version of the well-known ‘new better than used’ and ‘new worse than used’ ageing notions for random lifetimes within a specific cause of failure in the competing risks set up. Various properties of these new notions are studied, including the closure under time-dependent scale changes, as well as their connection to the hazard rate functions of the competing risks model. A data analytic example is finally provided.     

Some new results on the cumulative residual entropy

The residual entropy function is a relevant dynamic measure of uncertainty in reliability and survival studies. Recently, Rao et al. [2004. Cumulative residual entropy: a new measure of information. IEEE Transactions on Information Theory 50, 1220–1228] and Asadi and Zohrevand [2007. On the dynamic cumulative residual entropy. Journal of Statistical Planning and Inference 137, 1931–1941] define the cumulative residual entropy and the dynamic cumulative residual entropy, respectively, as some new measures of uncertainty. They study some properties and applications of these measures showing how the cumulative residual entropy and the dynamic cumulative residual entropy are connected with the mean residual life function. In this paper, we obtain some new results on these functions. We also define and study the dynamic cumulative past entropy function. Some results are given connecting these measures of a lifetime distribution and that of the associated weighted distribution.     

Proportional hazards model with quantile functions

Proportional Hazards Model (PHM) introduced by Cox (1972) is extensively studied in literature. In this paper, we study reliability properties of the PHM using quantile functions. Some special properties of the quantile function, which are not shared by distribution function are explored to study various properties of the PHM. We discuss ageing properties and stochastic orders for the PHM. The quantile-based dynamic cumulative Kullback-Leibler divergence of PHM is studied. Characterizations of some important quantile densities using PHM are also proved.     

A new three-parameter ageing distribution

We propose a new three-parameter ageing distribution called the Weibull-Poisson (WP) distribution, which generalizes the exponential-Poisson (EP) distribution introduced by Kus (2007). This new distribution has a more general form of failure rate (hazard rate) function. With appropriate choice of parameter values, it is able to model three ageing classes of life distributions including decreasing failure rate (DFR), increasing failure rate (IFR), and modified upside-down-bathtub (MUBT)-shaped failure rate. It thus provides an alternative to many existing life distributions. Various properties of this distribution are discussed and the estimation of the parameters is considered by the expectation maximization (EM) algorithm. Also, the asymptotic variance-covariance matrices of these estimates are obtained. Furthermore, some expressions for the Rènyi and Shannon entropies are given. Simulation studies are performed and experimental results are illustrated based on a real data set.     

A general family of NBU classes of life distributions

In this paper we introduce and study a family of new better than used ageing notions parameterized by a function h. The new better than used , the new better than used of a specified age and the new better than used in total time on test transform order , are special cases of this new family. We study some properties of the new family, and we give some applications of it in actuarial science, reliability theory and statistics.