Proportional Reversed Hazard Rate Research Articles

Stochastic comparisons of second-order statistics from dependent and heterogeneous modified proportional (reversed) hazard rates scale models

<abstract><p>In this paper, we investigate the problem of stochastically comparing the second-order statistics from dependent and heterogeneous samples following modified proportional hazard rates scale (MPHRS) and modified proportional reversed hazard rates scale (MPRHRS) models under Archimedean copula. We built some sufficient conditions for the usual stochastic order whenever the samples have different parameter vectors. Finally, some numerical examples were provided to illustrate the theoretical results.</p></abstract>

Relative Orderings of Modified Proportional Hazard Rate and Modified Proportional Reversed Hazard Rate Models

In this paper, we identify several relative ordering properties of the modified proportional hazard rate and modified proportional reversed hazard rate models. For this purpose, we use two well-known relative orderings, namely the relative hazard rate ordering and the relative reversed hazard rate ordering. The investigation is to see how a relative ordering between two possible base distributions for the response distributions in these models is preserved when the parameters of the underlying models are changed. We will give some examples to illustrate the results and the conditions under which they are obtained. Numerical simulation studies have also been provided to examine the examples presented.

Comparisons of coherent systems with two types of heterogeneous components having proportional reversed hazard rates

AbstractThe comparison of coherent systems in terms of stochastic orders is vital in reliability theory. While there is a considerable amount of literature devoted to comparing systems with homogeneous and independent components, real‐world systems often consist of heterogeneous components. Hence, this article aims to investigate systems with heterogeneous and independent components, as well as, those with heterogeneous and dependent components. For this purpose, we consider systems comprise of three components, which are of two different types of components, namely two components of type A and one component of type B. The system's lifetime distribution is represented using the failure signature when the components are independent, which is a function of the component's life distribution. However, when the components are dependent, the system's lifetime distribution is represented using copula and diagonal sections. Additionally, distorted distributions are utilized to enable distribution‐free stochastic comparisons. Using these representations, we compare systems with components having proportional reversed hazard rates, in three scenarios: (i) when components are heterogeneous and independent; (ii) when components are heterogeneous and dependent; and finally, (iii) comparing systems with homogeneous and independent components with those that have heterogeneous components. To illustrate the applicability of these results, we provide some examples and applications.

Cumulative α-Jensen–Shannon measure of divergence: Properties and applications

The problem of quantifying the distance between distributions arises in various fields, including cryptography, information theory, communication networks, machine learning, and data mining. In this article, the analogy with the cumulative Jensen–Shannon divergence, defined in Nguyen and Vreeken (2015), we propose a new divergence measure based on the cumulative distribution function and call it the cumulative α-Jensen–Shannon divergence, denoted by . Properties of are studied in detail, and also two upper bounds for are obtained. The simplified results under the proportional reversed hazard rate model are given. Various illustrative examples are analyzed.

Estimation of Reliability in Multicomponent Set-up when Stress and Strength are Non-identical

In the multicomponent stress–strength reliability literary work, though assumption of identical stress and strengths components may not be universally true but have been commonly considered. Herein, we consider a multicomponent stress–strength model having k identical strength components, say X_i,~i=1,2,...,k, which are exposed to a common random stress Y, so that Y, X_i,~i=1,2,...,k are independent. Two different generalized cases of estimation are considered; namely, (i) when the strength components follow the proportional hazard rate model and stress component follows the proportional reversed hazard rate model and (ii) when the strength components follow the proportional reversed hazard rate model and stress component follow the proportional hazard rate model. In each case, maximum likelihood and Bayesian approach using Markov Chain Monte Carlo method has been utilized to estimate the reliability of the system. Asymptotic confidence intervals are also constructed for the reliability functions in both the cases. Monte Carlo simulation study is exercised to compare the performance of the aforementioned estimators. A real data example is also explored to examine the utility of the paper.

HR and RHR orderings of extremes of dependent variables under Archimedean copula

In this article, we discuss sufficient conditions for the hazard rate order on smallest order statistics for variables with Archimedean survival copula and having modified proportional hazard rates (modified scale) model. We also establish the reversed hazard rate order on largest order statistics from variables with Archimedean copula and having modified proportional reversed hazard rates model. The results established here extend the recent results of Li and Li (2019).

Two-sample nonparametric test for proportional reversed hazards

In the last few decades, several works have been undertaken in the context of proportional reversed hazard rates (PRHR) but any specific statistical methodology for the PRHR hypothesis is absent in the literature. This paper proposes a two-sample nonparametric test based on two independent samples to verify the PRHR assumption. Based on a consistent U-statistic three statistical methodologies have been developed exploiting U-statistics theory, jackknife empirical likelihood and adjusted jackknife empirical likelihood method. A simulation study has been performed to assess the merit of the proposed test procedure. Finally, the proposed procedure is applied to a data set in the context of brain injury-related biomarkers and a data set related to Duchenne muscular dystrophy.

On dynamic cumulative past inaccuracy measure based on extropy

An alternate measure of extropy based on the cumulative distribution function rather than the probability density function of a random variable is called the cumulative past extropy (CPE). In this communication, the concept of CPE has been extended to cumulative past extropy inaccuracy (CPEI) and then to a dynamic version of it. A characterization problem for the proposed dynamic extropy inaccuracy measure has been studied under the proportional reversed hazard rate model. We also discuss the stochastic ordering of dynamic cumulative past extropy and certain results based on it. Some well-known lifetime distributions have been characterized using the proposed dynamic cumulative past extropy inaccuracy (DCPEI) measure. Three methods for nonparametric estimation of the DCPEI measure based on kernel and empirical estimators are proposed. Simulation results show that the kernel-based estimators of the DCPEI measure perform better than the empirical-based estimator under different sample sizes and times. A real data set is considered to show the behavior of the estimators in real cases.

Stress–strength reliability estimation in a system with p-type non-identical multicomponents from PRHR family based on records

In this article, we estimate stress-strength reliability of a system containing K components of type p when non-identical strength variates and common stress variate follow proportional reversed hazard rate (PRHR) family of distribution for lower records. The maximum likelihood estimator, an asymptotic distribution and logit-scale transformed asymptotic confidence intervals of the stress-strength reliability are studied for unknown and known θ. Further, the Bayes estimates of stress-strength reliability with squared error loss function are obtained for unknown and known θ. In the deficiency of the explicit forms of the Bayesian estimators, the Lindley's approximation method and Markov Chain Monte Carlo methods are followed. Further, the credible interval with the highest posterior density is obtained through Markov Chain Monte Carlo methods. The performance of the estimators is carried out through simulation technique. A presentation of two real data is done here to exhibit the application of the derived results.

Distortion representations of multivariate distributions

The univariate distorted distributions were introduced in risk theory to represent changes (distortions) in the expected distributions of some risks. Later, they were also applied to represent distributions of order statistics, coherent systems, proportional hazard rate and proportional reversed hazard rate models, etc. In this paper we extend this concept to the multivariate setup. We show that, in some cases, they are a valid alternative to the copula representation, especially when the marginal distributions may not be easily handled. Several examples illustrate the applications of such representations in statistical modeling. They include the study of paired (dependent) ordered data, joint residual lifetimes, order statistics and coherent systems

Varentropy of inactivity time of a random variable and its related applications

Abstract The information content of a random variable is of importance in the field of information theory. The mean and the variance of the information content of a random variable are called entropy and varentropy of this random variable, respectively. The varentropy of the inactivity time of a random variable is studied and termed by past varentropy in this paper. Reliability properties associated with the past varentropy and the reversed hazard rate function are discussed. Lower and upper bounds for the past varentropy are provided. The varentropy is applied for the proportional reversed hazard rate model. Also, asymptotic distribution for the information content of the inactivity time of a random variable is provided based on past entropy and varentropy.

Stochastic comparisons on extreme order statistics from observations associated by FGM copula

This study deals with the stochastic comparison of extreme order statistics from random samples with FGM copula or FGM survival copula. For general marginal distributions, several sufficient conditions on the usual stochastic order, the reversed hazard rate order, the hazard rate order, the increasing convex order and the increasing concave order on sample maximums or minimums are derived. Then, we apply these conditions to samples following proportional hazard rate model, proportional reversed hazard rate model, Weibull distributions and negative binomial distributions, and generalize some ordering results about samples having independent observations to the case with FGM copula or FGM survival copula. Two comparing results on the second-order statistics from proportional hazard samples are presented. Some numerical examples are provided to illustrate the theoretical results as well.

Generalized proportional reversed hazard rate distributions with application in medicine

Generalized proportional reversed hazard rate distributions with application in medicine

Prediction Based on Type-II Censored Coherent System Lifetime Data under a Proportional Reversed Hazard Rate Model

Prediction Based on Type-II Censored Coherent System Lifetime Data under a Proportional Reversed Hazard Rate Model

Comparisons of Parallel Systems with Components Having Proportional Reversed Hazard Rates and Starting Devices

In this paper, we consider stochastic comparisons of parallel systems with proportional reversed hazard rate (PRHR) distributed components equipped with starting devices. By considering parallel systems with two components that PRHR and starting devices, we prove the hazard rate and reversed hazard rate orders. These results are then generalized for such parallel systems with n components in terms of usual stochastic order. The establish results are illustrated with some examples.

Ordering Awad–Varma Entropy and Applications to Some Stochastic Models

We consider a generalization of Awad–Shannon entropy, namely Awad–Varma entropy, introduce a stochastic order on Awad–Varma residual entropy and study some properties of this order, like closure, reversed closure and preservation in some stochastic models (the proportional hazard rate model, the proportional reversed hazard rate model, the proportional odds model and the record values model).

Discussion about inaccuracy measure in information theory using co-copula and copula dual functions

Inaccuracy measure is an important measure in information theory, which have been considered recently by many researchers, so that various generalizations have been introduced for this measure. In this paper, two new inaccuracy measures using co-copula and dual of a copula in copula theory are introduced and their properties under specific conditions are investigated. Including, under the establishment of proportional reversed hazard rate model and proportional hazard rate model, we obtain bounds and inequalities for these two inaccuracy measures, and we show that the triangle inequality can also exist for both of these measures. Also, under the assumption of radial symmetry, we prove the equality of these two new inaccuracies. In addition, we obtain a characterization property using the equality of these two inaccuracy measures for radially symmetric distributions. We provide examples to evaluate the results. Finally, in supplementary material section, by introducing estimators for the introduced inaccuracy measures, we examine some of the results using simulation methods and provide an example with real data.

Ordering results of extreme order statistics from dependent and heterogeneous modified proportional (reversed) hazard variables

In this paper, we carry out stochastic comparisons on extreme order statistics (i.e. smallest and largest order statistics) from dependent and heterogeneous samples following modified proportional hazard rates (MPHR) and modified proportional reversed hazard rates (MPRHR) models. We build the usual stochastic order for sample minimums and maximums, and the hazard rate order on minimums of sample and the reversed hazard rate order on maximums of sample are also derived, respectively. Finally, some examples are given to illustrate the theoretical results.

Stochastic comparisons of largest-order statistics for proportional reversed hazard rate model and applications

AbstractWe investigate stochastic comparisons of parallel systems (corresponding to the largest-order statistics) with respect to the reversed hazard rate and likelihood ratio orders for the proportional reversed hazard rate (PRHR) model. As applications of the main results, we obtain the equivalent characterizations of stochastic comparisons with respect to the reversed hazard rate and likelihood rate orders for the exponentiated generalized gamma and exponentiated Pareto distributions. Our results recover and strengthen some recent results in the literature.

New results on stochastic comparisons of finite mixtures for some families of distributions

The classical finite mixture model is an effective tool to describe the lifetimes of the items existing in a random sample which are selected from some heterogeneous populations. This paper carries out stochastic comparisons between two classical finite mixture models in the sense of the usual stochastic order, when the subpopulations follow a wide class of distributions including the scale model, the proportional hazard rate model and the proportional reversed hazard rate model. Next, we consider the hazard rate and dispersive orders when the subpopulations follow the proportional hazard rate model and also the reversed hazard rate order in the case that the subpopulations follow the proportional reversed hazard rate model. Finally, the likelihood ratio order between two finite mixtures is characterized when the subpopulations belong to the transmuted-G model.