Mean Residual Life Order Research Articles

Some New Results on Stochastic Comparisons of Spacings of Generalized Order Statistics from One and Two Samples

Generalized order statistics (GOSs) are often adopted as a tool for providing a unified approach to several stochastic models dealing with ordered random variables. In this contribution, we first recall various useful results based on the notion of total positivity. Then, some stochastic comparisons between spacings of GOSs from one sample, as well as two samples, are developed under the more general assumptions on the parameters of the model. Specifically, the given results deal with the likelihood ratio order, the hazard rate order and the mean residual life order. Finally, an application is demonstrated for sequential systems.

Preservation of mean inactivity time ordering for coherent systems

AbstractPreservation of stochastic orders through the system signature has captured the attention of researchers in recent years. Signature-based comparisons have been made for the usual stochastic order, hazard rate order, and likelihood ratio orders. However, for the mean residual life (MRL) order, it has recently been proved that the preservation result does not hold true in general, but rather holds for a particular class of distributions. In this paper, we study whether or not a similar preservation result holds for the mean inactivity time (MIT) order. We prove that the MIT order is not preserved from signatures to system lifetimes with independent and identically distributed (i.i.d.) components, but holds for special classes of distributions. The relationship between these classes and the order statistics is also highlighted. Furthermore, the distribution-free comparison of the performance of coherent systems with dependent and identically distributed (d.i.d.) components is studied under the MIT ordering, using diagonal-dependent copulas and distorted distributions.

Stochastic Comparisons of Largest-Order Statistics and Ranges from Marshall–Olkin Bivariate Exponential and Independent Exponential Variables

Sample range and the associated functions such as survival function and mean residual life function have found many important applications in the reliability field. In this work, we establish some results that are in two different directions. In the first part, we establish some conditions for comparing the largest-order statistics (in the sense of mean residual life order) arising from bivariate Marshall–Olkin exponential distribution. Then, in the second part, we present some sufficient conditions for comparing sample ranges (in the sense of usual stochastic order and reversed hazard rate order) arising from independent exponential random variables.

MRL ordering of largest order statistics from heterogeneous scale variables

ABSTRACT For comparing largest order statistics from independent heterogeneous non-negative scale variables in the mean residual life order, a new framework is introduced here. This framework can be viewed as a generalization of the well-known multiple-outlier scale model, and it additionally includes the situation in which all the random variables are heterogeneous. We also find some sufficient conditions for comparing the largest order statistics, one with complete heterogeneous scale parameters and another with homogeneous scale parameters, in the mean residual life order. As examples of the obtained results, generalized gamma, generalized beta of the second kind, power-generalized Weibull, and half-normal distributions are all presented. The findings of this work generalize and also reinforce some of the existing results in this direction.

Comparison of Coherent Systems with Dependent Components Based on ‌‎the ‎Reversed Mean Residual Life Order

Comparison of Coherent Systems with Dependent Components Based on ‌‎the ‎Reversed Mean Residual Life Order

Mean residual life order among largest order statistics arising from resilience-scale models with reduced scale parameters

In this paper, we identify some conditions to compare the largest order statistics from resilience-scale models with reduced scale parameters in the sense of mean residual life order. As an example of the established result, the exponentiated generalized gamma distribution is examined. Also, for the special case of the scale model, power-generalized Weibull and half-normal distributions are investigated.

Jackknife empirical likelihood ratio test for testing mean residual life and mean past life ordering

We propose nonparametric tests for testing mean residual life and mean past life ordering. Asymptotic properties of the proposed test statistics are studied. We also propose Jackknife Empirical Likelihood (JEL) ratio tests for testing mean residual life and mean past life ordering. Some numerical results are presented to evaluate the performance of the proposed tests by comparing them with few competing tests available in the literature. Finally, we illustrate the proposed test procedures using real data sets.

Test for harmonic mean residual life function: A goodness of fit approach

A family of goodness of fit tests is proposed involving dominance in harmonic mean residual life (HMRL) order. We develop a test procedure based on a weighted integral approach. The test statistic proposed has been shown to be asymptotically normal — in addition the consistency of the proposed test has been established. Furthermore, the performance of the test has been assessed by means of a simulation study. Finally, we apply our test to a real-life dataset for purposes of illustration.

A New Discrete Mean Residual Life Order and its Reliability Applications

A New Discrete Mean Residual Life Order and its Reliability Applications

Comparing residual lifetimes and inactivity times of dependent random variables

In this paper, sufficient conditions are presented for comparing the residual lifetimes and inactivity times of dependent random variables at different fixed times, with respect to hazard rate, mean residual life and likelihood ratio orders. The stochastic comparisons of residual lifetimes at different random times are discussed next. Several examples are presented to illustrate the established results.

Logit Truncated-Exponential Skew-Logistic Distribution with Properties and Applications

In recent years, bounded distributions have attracted extensive attention. At the same time, various areas involve bounded interval data, such as proportion and ratio. In this paper, we propose a new bounded model, named logistic Truncated exponential skew logistic distribution. Some basic statistical properties of the proposed distribution are studied, including moments, mean residual life function, Renyi entropy, mean deviation, order statistics, exponential family, and quantile function. The maximum likelihood method is used to estimate the unknown parameters of the proposed distribution. More importantly, the applications to three real data sets mainly from the field of engineering science prove that the logistic Truncated exponential skew logistic distribution fits better than other bounded distributions.

Optimal prevention of large risks with two types of claims

In this paper, we propose and study a risk model with two types of claims in which the insurer may invest into a prevention plan which decreases the intensity of large claims without impacting the small claims. We identify a necessary and sufficient condition for insurers to use prevention if there is no surplus. If, in addition, the severity of large claims dominates that of small claims by the harmonic mean residual life (HMRL) order, insurers invest more in prevention in the presence of a surplus. Finally, we characterize the asymptotic optimal prevention strategy when the initial surplus tends to infinity in the two main cases where both claim types are light-tailed and where one of them is light-tailed and the other one is heavy-tailed.

Relative stochastic orders of weighted frailty models

ABSTRACT In this paper, relative stochastic comparisons of weighted frailty models with respect to relative hazard rate and relative mean residual life orders are considered. Some closure properties of the model in two relative stochastic orders sense are presented. Under some appropriate assumptions, we show how the variation of the frailty variable and the variation of the baseline variable translate into these two relative stochastic orders between the overall population variables.

Further results on residual life and inactivity time at random time

Stochastic comparisons of residual life (inactivity time) of a random variable X at random time Y (RLRT/ITRT) with respect to likelihood ratio, (reversed) hazard rate, mean residual life and variance residual life orders have been investigated in the literature. In the present note, we provide some more stochastic ordering results for RLRT and ITRT based on reversed hazard rate, mean inactivity time and variance inactivity time orders. We also study various properties of ITRT based on DRHR, IMIT and IVIT classes of life distributions.

Preservation of the mean residual life order for coherent and mixed systems

AbstractThe signature of a coherent system has been studied extensively in the recent literature. Signatures are particularly useful in the comparison of coherent or mixed systems under a variety of stochastic orderings. Also, certain signature-based closure and preservation theorems have been established. For example, it is now well known that certain stochastic orderings are preserved from signatures to system lifetimes when components have independent and identical distributions. This applies to the likelihood ratio order, the hazard rate order, and the stochastic order. The point of departure of the present paper is the question of whether or not a similar preservation result will hold for the mean residual life order. A counterexample is provided which shows that the answer is negative. Classes of distributions for the component lifetimes for which the latter implication holds are then derived. Connections to the theory of order statistics are also considered.

Some unified results on stochastic properties of residual lifetimes at random times

The residual life of a random variable $X$ at random time $\Theta$ is defined to be a random variable $X_{\Theta}$ having the same distribution as the conditional distribution of $X-\Theta$ given $X>\Theta$ (denoted by $X_{\Theta}=(X-\Theta|X>\Theta)$). Let $(X,\Theta_{1})$ and $(Y,\Theta_{2})$ be two pairs of jointly distributed random variables, where $X$ and $\Theta_{1}$ (and, $Y$ and $\Theta_{2}$) are not necessarily independent. In this paper, we compare random variables $X_{\Theta_{1}}$ and $Y_{\Theta_{2}}$ by providing sufficient conditions under which $X_{\Theta_{1}}$ and $Y_{\Theta_{2}}$ are stochastically ordered with respect to various stochastic orderings. These comparisons have been made with respect to hazard rate, likelihood ratio and mean residual life orders. We also study various ageing properties of random variable $X_{\Theta_{1}}$. By considering this generalized model, we generalize and unify several results in the literature on stochastic properties of residual lifetimes at random times. Some examples to illustrate the application of the results derived in the paper are also presented.

Stochastic comparison of parallel systems with heterogeneous exponential components

Abstract In this paper we provide a sufficient condition for mean residual life ordering of parallel systems with n ≥ 3 heterogeneous exponential components.

Answer to an open problem on mean residual life ordering of parallel systems under multiple-outlier exponential models

In a comprehensive survey on stochastic comparison, Balakrishnan and Zhao (2013) proposed an open problem on mean residual life ordering between two parallel systems under multiple-outlier exponential models. This paper provides an affirmative answer to this open problem.

Stochastic Properties of Inactivity Lifetimes at Random Times: Some Unified Results

SYNOPTIC ABSTRACTIn this article, we consider the inactivity times and of two random variables X and Y at random times Θ1 and Θ2, respectively, and make stochastic comparisons of and with respect to likelihood ratio, hazard rate and mean residual life orders. Here, X and Θ1 (and Y and Θ2) may be dependent random variables. We also study various ageing properties of random variable . The results derived here unify and extend several existing results. An application of the results derived in this paper is also illustrated.

Comparisons of smallest order statistics from Pareto distributions with different scale and shape parameters

We derive conditions for stochastic, hazard rate, likelihood ratio, reversed hazard rate, increasing convex and mean residual life orderings of Pareto distributions with different shape and scale parameters. A real data application of the conditions is presented.