Shock models leading to non-monotonic ageing classes of life distributions

We consider the life distribution of a device subject to a sequence of shocks occurring randomly in time according to a homogeneous Poisson process. Under appropriate conditions on the probability of surviving a given number of shocks, it is shown that fundamental non-monotonic ageing classes such as Bathtub Failure Rate (BFR) and New Worse then Better than Used in Expectation (NWBUE) arise from the shock model under consideration.     

Moment inequalities derived from comparing life with its equilibrium form

A life is characterized by a nonnegative random variable. Associated with life, two notions are interesting in applications such as biomedical research, engineering and statistics. These are the notions of random remaining life at a certain age and its equilibrium (stationary) limit as the age tends to infinity. In the current investigation, comparisons between these three notions both stochastically and in expectation are discussed where moments inequalities based on these comparisons are presented. These inequalities are then used to develop testing the hypothesis of exponentiality against orderings based on these comparisons.     

Moments inequalities of aging families of distributions with hypotheses testing applications

Nonparametric families of aging distributions have been the subject of investigation for more than three decades. Both probabilistic and statistical properties of these distributions were studied for such families as “increasing failure rate”, “new better than used”, “new better than used in expectation”, and “harmonic new better than used in expectation”. In the present work, moments inequalities are derived for the above-mentioned four families that demonstrate that if the mean life is finite for any of them then all higher-order moments exist. Next, based on these inequalities, new testing procedures for exponentiality against any one of the above classes are introduced and studied showing that they are simpler than most earlier ones and hold high relative efficiency for some commonly used alternatives.     

TESTING NEW BETTER THAN RENEWAL USED LIFE DISTRIBUTIONS BASED ON U-TEST

In this paper, the problem of testing exponentiality against new better (worse) than renewal used in expectation is investigated and similarly for the case of nuharmonic new better than renewal used in expectation. For each of these two aging properties, a nonparametric procedure (U-statistic) is presented. Selected critical values are tabulated for sample sizes n = 5(1)30(10)50. The Pitman asymptotic relative efficiency to the test relative to other classes are studied. A real example is given to elucidate the use of the proposed test statistics for the reliability analysis.     

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.     

A new test for decreasing mean residual lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. One model of aging is the decreasing mean residual life (DMRL): M is decreasing (non increasing) in time. It vastly generalizes the more stringent model of increasing failure rate (IFR). The exponential distribution lies at the boundary of both of these classes. There is a large literature on testing exponentiality against DMRL alternatives which are all of the integral type. Because most parametric families of DMRL distributions are IFR, their relative merits have been compared only at some IFR alternatives. We introduce a new Kolmogorov–Smirnov type sup-test and derive its asymptotic properties. We compare the powers of this test with some integral tests by simulations using a class of DMRL, but not IFR alternatives, as well as some popular IFR alternatives. The results show that the sup-test is much more powerful than the integral tests in all cases.     

Recent tests of exponentiality against IFR alternatives: a survey

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 (which essentially means no aging) against positive aging, which is captured by the class of increasing failure rate alternatives. Recent tests are discussed and compared. The empirical size of the tests is obtained by simulation. Power computations, using simulations, are done for each test procedure. These comparisons are done both for small and large sample sizes. Suggestions are made regarding the choice of the test when a particular alternative is suspected.     

A review of tests for exponentiality with Monte Carlo comparisons

In this paper, 91 different tests for exponentiality are reviewed. Some of the tests are universally consistent while others are against some special classes of life distributions. Power performances of 40 of these different tests for exponentiality of datasets are compared through extensive Monte Carlo simulations. The comparisons are conducted for different sample sizes of 10, 25, 50 and 100 for different groups of distributions according to the shape of their hazard functions at 5 percent level of significance. Also, the techniques are applied to two real-world datasets and a measure of power is employed for the comparison of the tests. The results show that some tests which are very good under one group of alternative distributions are not so under another group. Also, some tests maintained relatively high power over all the groups of alternative distributions studied while some others maintained poor power performances over all the groups of alternative distributions. Again, the result obtained from real-world datasets agree completely with those of the simulation studies.KEYWORDS: Classes of life distributions, empirical power of a test, exponentiality, goodness-of-fit test, Monte Carlo simulationSubject Classifications: 62E10, 62E20, 62F03     

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.     

Tests for exponentiality against NBUE alternatives: a Monte Carlo comparison

Testing of various classes of life distributions has been addressed in the literature for more than 45 years. In this paper, we consider the problem of testing exponentiality (which essentially implies no ageing) against positive ageing which is captured by the fairly large class of new better than used in expectation (NBUE) distributions. These tests of exponentiality against NBUE alternatives are discussed and compared. The empirical size of the tests is obtained by simulations. Power comparisons for different popular alternatives are done using Monte Carlo simulations. These comparisons are made for both small and large sample sizes. The paper concludes with a discussion in which suggestions are made regarding the choices of the test when a particular alternative is suspected.     

An L-statistic approach to a test of exponentiality against IFR alternatives

In this note we consider the problem of testing exponentiality against IFR alternatives. A measure of deviation from exponentiality is developed and a test statistic constructed on the basis of this measure. It is shown that the test statistic is an L-statistic. The asymptotic as well as the exact distribution of the test statistic is obtained and the test is shown to be consistent.     

A New Test for New Better Than Used in Expectation Lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. A popular nonparametric model of aging is new better than used in expectation (NBUE), when M(t) ? M(0) for all t ? 0. The exponential distribution lies at the boundary. There is a large literature on testing exponentiality against NBUE alternatives. However, comparisons of tests have been made only for alternatives much stronger than NBUE. We show that a new Kolmogorov-Smirnov type test is much more powerful than its competitors in most cases.     

Testing exponentiality against L-distributions

In this paper we propose a test statistic for testing exponentiality versus L-class of life distributions. This test is based on an estimate of a functional of the cdf which discriminates between the exponential and L-family.     

A family of tests for exponentiality against IFR alternatives

In this paper we consider the problem of testing exponentiality against IFR alternatives. A measure of deviation from exponentiality is developed and a class of test statistics are constructed on the basis of this measure. It is shown that the test statistic is an L-statistic. The asymptotic as well as the exact distributions of the test statistics are obtained and the test statistics are proved to be consistent. The Pitman efficiency has also been studied.     

Testing dominance of biometric functions

The hazard rate (HR) and mean residual lifetime are two of the most practical and best-known functions in biometry, reliability, statistics and life testing. Recently, the reversed HR function is found to have interesting properties useful in additional areas such as censored data and forensic science. For these three biometric functions, we propose testing methods that they take on a known functional form against that they dominate or are dominated by this known form. This goodness-of-fit-type testing is wider in applications and more interesting than the long-standing testing procedures for exponentiality against the monotonicity of these functions or even the change point problems. This is so since we can test against any choice of the survival distribution and not just exponentiality. For this general testing, we present easy to implement tests and generalize them into classes of statistics that could lead to more powerful and efficient testing.     

Reversed hazard rate order of equilibrium distributions and a related aging notion

This paper deals with preservation of the reversed hazard rate order between equilibrium random variables under formations of some reliability structures. We further investigate a new aging notion based upon the reversed hazard rate order between a random life and its equilibrium version. A nonparametric method is developed to test the exponentiality against such a strict aging property, some numerical results are presented as well. This research was supported by the Program for New Century Excellent Talent in University, China.     

Testing for NBUL using goodness of fit approach with applications

Most of the approaches suggested during the last decades for solving life testing problems are markedly different from those used in the related but wider area of goodness of fit problems. In this paper, it is demonstrate that applying the goodness of fit approach makes sense also for solving life testing problems and that result in simpler procedures that are asymptotically equivalent or better than standard ones. They may have superior finite sample behavior. A U-statistic is obtained for testing exponentiality against the new better than used in the Laplace transform order aging class (NBUL). The pitman asymptotic efficiency of (NBUL) is studied. The power and the critical values of the proposed statistic are calculated. Real examples are given to elucidate the use of the proposed test statistic in the reliability analysis. Finally, the problem with the right censored data is also handled with some applications.     

Goodness-of-fit tests for exponentiality based on a loss-of-memory type functional equation

Three goodness-of-fit tests for exponentiality based on the functional equation characterization 1?F(2x)=[1?F(x)]² for every x?0 are proposed and shown to compare well to several popular tests against common alternative cdf's. Small-sample critical values for α=0.10,0.05 are developed for the two superior test statistics and the asymptotic null-distributions are characterized.     

A new aging concept derived from the increasing convex ordering

In this paper, a comparison between the life distribution of a new unit with that of a used unit in the increasing convex order is made leading to a new class of life distributions which we call “new better than used in convex ordering of second order”. This class includes as subclasses the NBU and the NBUC and is a subclass of the NBUCA class. Preservation properties under convolution, random maxima, mixing and formation of coherent structures are established. Stochastic comparisons of the excess lifetime when the inter-arrival times belong to the NBUC(2) class are developed. Some applications of Poisson shock models and a test of exponentiality against NBUC(2) alternative are presented.     

A new test for exponentiality against HNBUE alternatives

Abstract

Two families of test statistics for testing the null hypothesis of exponentiality against Harmonic New Better than Used in Expectation (HNBUE) alternatives are proposed. Asymptotic distributions of the test statistics are derived under the null and alternative hypotheses and the consistency of the tests established. Comparison with competing tests are made in terms of Pitman Asymptotic Relative Efficiency (PARE). Simulation studies have been carried out to assess the performance of the tests. Finally, the test has been applied to three real life data sets described in Proschan, Susarla and Van Ryzin and Engelhardt, Bain and Wright.