Decreasing Mean Residual Life Research Articles

On some properties of distributions possessing a bathtub-shaped failure rate average

AbstractThe life distribution of a device subject to shocks governed by a homogeneous Poisson process is shown to have a bathtub failure rate average (BFRA) when the probabilities $\bar{P}_k$ of surviving k shocks possess the corresponding discrete property. We prove closure under the formation of weak limits for BFRA distributions and explore related moment convergence issues within the BFRA family. Similar results for increasing and decreasing failure rate average distributions are obtained either independently or as consequences of our results. We also establish some results outlining the positions of various non-monotonic ageing classes such as bathtub failure rate, increasing initially then decreasing mean residual life, new worse then better than used in expectation, and increasing initially then decreasing mean time to failure in the hierarchy. Finally, an open problem is posed and a partial solution provided.

The Power Zeghdoudi Distribution: Properties, Estimation, and Applications to Real Right-Censored Data

A new two-parameter power Zeghdoudi distribution (PZD) is suggested as a modification of the Zeghdoudi distribution using the power transformation method. As a result, the PZD may have increasing, decreasing, and unimodal probability density function and decreasing mean residual life function. In addition, other properties are presented, such as moments, order statistics, reliability measures, Bonferroni and Lorenz curves, Gini index, stochastic ordering, mean and median deviations, and quantile function. Following this, a section is devoted to the related model parameters which are estimated using the maximum likelihood estimation method, the weighted least squares and least squares methods, the maximum product of spacing method, the Cramer–von Mises method, and the right-tail and left-tail Anderson–Darling methods, and the Nikulin–Rao–Robson test statistic is considered. A simulation study is conducted to assess these methods and to investigate the distribution properties with right-censored data. The applicability of the proposed model is studied based on three real data sets of failure times, bladder cancer patients, and glass fiber data with a comparison with such competitors as the gamma, xgamma, Lomax, Darna, power Darna, power Lindley, and exponentiated power Lindley models. According to several established criteria, the comparative findings are overwhelmingly favorable to the suggested model.

Markov's inequality: Sharpness, renewal theory, finite samples, reliability theory

We examine Markov's inequality in the light of renewal theory and reliability theory. Suppose the non negative random variable (rv) X has cumulative distribution function (cdf) F with survival function and left-continuous version of the survival function We determine the points, if any, such that and We offer an alternative proof of Markov's inequality by observing that, if some collection of events exists such that then because equals a probability, it must satisfy which is equivalent to Markov's inequality. We choose events connected to stationary renewal processes. When we know only the sample size n and the sample average of n non negative observations we establish an upper bound on the left-continuous version of the empirical survival function that improves Markov's inequality. We show that an upper bound of Markov type for the survival function is sharp when F is “new better than used in expectation” (NBUE) or has “decreasing mean residual life” (DMRL).

Some New Bounds for Moment Generating Functions of Various Life Distributions Using Mean Residual Life Functions

In this paper, we present bounds for moment generating functions of certain life distributions, including the decreasing mean residual life class. We also give bounds for some other much larger classes of life distributions. Most bounds are based on the mean residual life functions. These bounds are compared to some other bounds in the literature. The methods used can also find bounds for other functions of the lifetime random variable. Some applications to bounding tail probabilities of lifetime distributions are given.

SHARP BOUNDS FOR SURVIVAL PROBABILITY WHEN AGEING IS NOT MONOTONE

We exploit a novel bounding argument to obtain sharp bounds for survival functions belonging to the Increasing initially then Decreasing Mean Residual Life (IDMRL) class introduced by Guess, Hollander and Proschan (1986) [8]. The bounds obtained are in terms of the mean, change point and pinnacle of the mean residual life function. The bounds for the monotonic ageing classes Decreasing Mean Residual Life (DMRL) and Increasing Mean Residual Life (IMRL) are obtained as special cases. Discussions on the bounds as well as two concrete illustrative examples are included.

On the Properties of the UBAC(2) Class of Life Distributions

Abstract This paper provided some properties of a new class of life distributions that are denoted as used better than aged in an increasing concave ordering class (UBAC(2)). This class includes many classes of life distributions like used better than aged (UBA), used better than aged in convex ordering (UBAC), used better than aged in expectation (UBAE) and decreasing mean residual life (DMRL). The considered properties included the moment inequalities, moment generating function behavior, and testing hypotheses about the survival function of this class. The distribution of the proposed statistic was investigated via simulation. We used real data as an application of the proposed test statistic.

Sharp bounds on DMRL and IMRL classes of life distributions with specified mean

We obtain sharp upper and lower bounds for a reliability function with decreasing mean residual life (DMRL), in terms of its mean. The constructive proofs establish that the bounds are sharp. We also provide bounds for the IMRL class.

Sharp bounds for exponential approximations under a hazard rate upper bound

Consider an absolutely continuous distribution on [0, ∞) with finite meanμand hazard rate functionh(t) ≤bfor allt. Forbμclose to 1, we would expectFto be approximately exponential. In this paper we obtain sharp bounds for the Kolmogorov distance betweenFand an exponential distribution with meanμ, as well as betweenFand an exponential distribution with failure rateb. We apply these bounds to several examples. Applications are presented to geometric convolutions, birth and death processes, first-passage times, and to decreasing mean residual life distributions.

Sharp bounds for exponential approximations under a hazard rate upper bound

Consider an absolutely continuous distribution on [0, ∞) with finite mean μ and hazard rate function h(t) ≤ b for all t. For bμ close to 1, we would expect F to be approximately exponential. In this paper we obtain sharp bounds for the Kolmogorov distance between F and an exponential distribution with mean μ, as well as between F and an exponential distribution with failure rate b. We apply these bounds to several examples. Applications are presented to geometric convolutions, birth and death processes, first-passage times, and to decreasing mean residual life distributions.

Improved Asymptotics of a Decreasing Mean Residual Life Estimator

The mean residual life of a life distribution, X, with a finite mean is defined by M(t) = E[X − t|X > t] for t ⩾ 0. Kochar et al. (2000) provided an estimator of M when it is assumed to be decreasing. They showed that its asymptotic distribution was the same as that of the empirical estimate, but only under very stringent analytic and distributional assumptions. We provide a more general asymptotic theory, and under much weaker conditions. We also provide improved asymptotic confidence bands.

Nonparametric Testing for More DMRL Ordering

A test proposed for testing the null hypothesis that two life distributions are equal against the alternative that one is more Decreasing Mean Residual Life (DMRL) ordering. The asymptotic normality of the proposed test statistic was also established. The empirical size and empirical power of the proposed test are simulated for some specific families of distributions like beta and Weibull that are ordered with respect to more DMRL order. Finally, we apply our test to some real data sets.

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.

A family of test statistics for trend change in mean residual life with unknown turning point using censored data

In this paper, we develop a family of test statistics for testing exponentiality against increasing then decreasing mean residual life alternatives to accommodate the randomly censored data when neither the change point nor the proportion is known. We establish the asymptotic null distribution of test statistics. Monte Carlo simulations are conducted to investigate the speed of convergence of test statistics to the asymptotic null distribution and study the performance of test statistics by the power of tests.

New multivariate orderings based on conditional distributions

In this paper, we propose and study new multivariate extensions of the dispersive, right‐spread, decreasing mean residual life and new better than used in expectation univariate orders. These new orders are based on the comparison of univariate marginal distributions conditional on survival data for the rest of the components. Relationships among multivariate orders and applications to some multivariate random vectors are also provided. Copyright © 2011 John Wiley & Sons, Ltd.

Nonparametric Estimation of Decreasing Mean Residual Life With Type II Censored Data

In this paper, a nonparametric method is proposed for the estimation of decreasing mean residual life with type II censored data. This method is based on the comparison between two estimators of the reliability function: the Kaplan-Meier estimator, and an estimator derived from the empirical MRL function. Simulation results indicate that the new approach is able to give good performance, and can outperform some existing parametric methods when censoring is heavy.

Characterizations of Aging Classes in Terms of Spacings Between Record Values

Nonparametric classes of life distributions, such as the increasing likelihood ratio, increasing failure rate, decreasing mean residual life, new better than used, new better than used in the convex order, and new better than used in expectation classes as well as their duals, are characterized in terms of properties of the spacings between record values. A generalization of the usual record value sequence is needed.

A New Test Procedure for Decreasing Mean Residual Life

A new test statistic for testing the strict DMRL property of life distribution is developed. The asymptotic normality is established and the comparison between the test proposed and some other related ones in literature is conducted through evaluating the Pitman's asymptotic relative efficiency. Edge-worth expansion is also employed to improve the accuracy of the convergence rate of the test statistic. Some numerical results are presented as well to demonstrate the performance and the asymptotic normality of the new testing procedure.

SOME RESULTS ABOUT MIT ORDER AND IMIT CLASS OF LIFE DISTRIBUTIONS

We investigate some new properties of mean inactivity time (MIT) order and increasing MIT (IMIT) class of life distributions. The preservation property of MIT order under increasing and concave transformations, reversed preservation properties of MIT order, and IMIT class of life distributions under the taking of maximum are developed. Based on the residual life at a random time and the excess lifetime in a renewal process, stochastic comparisons of both IMIT and decreasing mean residual life distributions are conducted as well.

On the Inter-Relationships Between Some Classes of Life Distributions

It is well lmowu that if F is IFR, theu it is also DM RI; however, the converse is not true, in general. Kupka and Loo (1989) proved that a distribution with convex decreasing mean residual life (CDMRL) function has an Increasing Failure Rate (IFR) distribution. In this paper, we show that this condition (viz., convexity of DMRL) is not nenessary for F to be IF R. It is observed that contrary to the monotonic classes, in the non-monotonic classes, F is BFR does not imply that F is IDMRL. In this paper, we derive a result that gives an additional condition required for a BF R distribution to be an I DM RL distribution.

Stop-Loss Transformierte eines höheren Grades und stochastische Ordnungen - (I) Theorie

The higher degree stop-loss transforms and their logarithmic derivatives, called higher degree stop-loss rate functions, are studied to get insight into the hierarchical theory of the higher degree stop-loss orders and related stochastic orders. Based on differential-integral recursive relationships, we derive in a simple way two characterization results by Gupta and Gupta (1983), which state that the higher degree stop-loss transforms and the higher degree stop-loss rate functions uniquely determine a distribution function. Classes ISLR (n) of distributions with an increasing stop-loss rate function of degree n are considered, and it is shown that ISLR (n) implies ISLR (n+1). This result generalizes the well-known fact by Bryson and Siddiqui (1969) that a distribution with an increasing failure rate has necessarily a decreasing mean residual life. Necessary and sufficient conditions, which guarantee that ISLR (n+1) implies ISLR (n), are formulated. Using notions of higher degree stop-loss rate order and higher degree stop-loss rate dangerousness order, sufficient conditions for a higher degree stop-loss order relation are established. Two new sign change characterizations of the higher degree stop-loss order by means of higher degree stop-loss transforms and higher degree stop-loss rate functions are derived. Applications in actuarial mathematics follow in part (II) of the present work.