Hazard Rate Order Research Articles

The Hybrid Odd Exponential-Exponential Distribution: Statistical Properties and Applications

In this paper, we study a new statistical distribution called the hybrid odd exponential-exponential distribution (HOE-E distribution for short) we the aim of applying it to real data sets. This new distribution appears as a sub-model of the larger HOEE-Φ family in (Mahdi et al., 2024). We explore many mathematical and statistical properties of this distribution such as the series expansion of the PDF, the quantile function, moments and their generating function, incomplete moments, the shape of the PDF, and the shape of the hazard rate function, order statistics, and parameter estimation using the maximum likelihood estimation. We use this distribution to model a plasma concentration data set. Moreover, this we show that the suggested HOE-E distribution outperformance many other distributions based on various statistical criteria like AIC, CAIC, BIC, HQIC, W, and A.

Stochastic comparison of parallel systems with heterogeneous dependent exponential components

Let X=(X1,…,Xn) and Y=(Y1,…,Yn) be two random vectors with common Archimedean copula with generator function ϕ, where, for i=1,…,n, Xi is an exponential random variable with hazard rate λi and Yi is an exponential random variable with hazard rate λ. In this paper we prove that under some sufficient conditions on the function ϕ, the largest order statistic corresponding to X is larger than that of Y according to the dispersive ordering and hazard rate ordering. The new results generalized the results in Dykstra et al. (1997) and Khaledi and Kochar (2000). We show that the new results can be applied to some well known Archimedean copulas.

Ordering results between two multiple-outlier finite [formula omitted]-mixtures

In this paper, we have obtained sufficient conditions for comparing two multiple-outlier (M-O) finite δ-mixtures based on the usual stochastic order and reversed hazard rate order. We have assumed a general parametric family of distributions for the subpopulations. Many distributions satisfying baseline-related conditions in the established results have also been provided as examples.

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.

Orderings of extremes among dependent extended Weibull random variables

Abstract In this work, we consider two sets of dependent variables $\{X_{1},\ldots,X_{n}\}$ and $\{Y_{1},\ldots,Y_{n}\}$ , where $X_{i}\sim EW(\alpha_{i},\lambda_{i},k_{i})$ and $Y_{i}\sim EW(\beta_{i},\mu_{i},l_{i})$ , for $i=1,\ldots, n$ , which are coupled by Archimedean copulas having different generators. We then establish different inequalities between two extremes, namely, $X_{1:n}$ and $Y_{1:n}$ and $X_{n:n}$ and $Y_{n:n}$ , in terms of the usual stochastic, star, Lorenz, hazard rate, reversed hazard rate and dispersive orders. Several examples and counterexamples are presented for illustrating all the results established here. Some of the results here extend the existing results of [5] (Barmalzan, G., Ayat, S.M., Balakrishnan, N., & Roozegar, R. (2020). Stochastic comparisons of series and parallel systems with dependent heterogeneous extended exponential components under Archimedean copula. Journal of Computational and Applied Mathematics380: Article No. 112965).

Stochastic comparison results between two finite mixture models with generalized Weibull distributed components

In this paper, we establish sufficient conditions for stochastic comparisons of two finite mixture models (FMMs) with respect to the usual stochastic order, hazard rate order, and likelihood ratio order when the mixing components have generalized Weibull family of distributions. The established (sufficient) conditions are mainly based on the majorization order, weak supermajorization order, and weak submajorization order. The stochastic comparisons are studied when there is heterogeneity in one (model) parameter, and then in two parameters (model parameter and mixing proportion). Further, the concept of unordered majorization order is employed to establish the usual stochastic order between two FMMs. To illustrate the theoretical results established here, several numerical examples and counterexamples are presented. Finally, we have generalized some of the results to the case of τ-mixture models.

On Some Stochastic Ordering Comparisons for Renewal Processes

After a brief introduction to ordinary renewal process, we have defined compound renewal process generated by an ordinary renewal process followed by the distributional properties of the corresponding random variables. A few results are obtained by comparing independent renewal processes with respect to several stochastic orderings between the generating inter-arrival time random variables, like, stochastic order, hazard rate order, likelihood ratio order and variability order, as well as some ageing classes of the generating random variables. Some numerical illustrations are given. The results obtained here appear to be new.

U-statistics Based Tests for Marginal Hazard Rate Orderings of Two Dependent Variables

U-statistics Based Tests for Marginal Hazard Rate Orderings of Two Dependent Variables

Stochastic Orderings between Two Finite Mixtures with Inverted-Kumaraswamy Distributed Components

In this paper, we consider two finite mixture models (FMMs) with inverted-Kumaraswamy distributed components’ lifetimes. Several stochastic ordering results between the FMMs are obtained. Mainly, we focus on three different cases in terms of the heterogeneity of parameters. The usual stochastic order between the FMMs is established when heterogeneity presents in one parameter as well as two parameters. In addition, we also study ageing faster order in terms of the reversed hazard rate between two FMMs when heterogeneity is in two parameters. For the case of heterogeneity in three parameters, we obtain the comparison results based on reversed hazard rate and likelihood ratio orders. The theoretical developments are illustrated using several examples and counterexamples.

When Yield Is Not the Only Supply Uncertainty: Newsvendor Model of a Trade Agent

Problem definition: We study the procurement decisions of a trade agent: The agent chooses a bid (unit price to pay) to procure the goods available from seller(s). If the agent wins the bid, the supply is used to meet the buyer’s demand. Methodology/results: The trade agent’s single-period, single-product problem is a new type of newsvendor problem. We analyze the agent’s optimal bid for a seller with yield uncertainty. We show that the bid outcome distribution needs to satisfy an easy-to-check condition but no conditions on the yield distribution are needed for a unique optimal bid to exist. We also show that the expected sales-to-supply ratio that measures scarcity affects the optimal bid. We investigate the monotonicity of the optimal bid with respect to economic parameters, demand, and distributions of bid outcome and yield. The agent’s problem with multiple sellers is also a novel newsvendor network problem. For the two-seller case, we show when diversification is optimal for the agent. We show that working with both sellers may not always be optimal despite the opportunity for risk pooling and bidding only at the unreliable seller may be optimal even when the other seller is reliable. Managerial implications: We make the following recommendations for the agent: (i) bid at a seller only when the expected sales-to-supply ratio for a seller is higher than the critical ratio, considering the agent’s cost of underage and overage, (ii) increase the bid if the bid outcome distribution increases in the reversed hazard rate order, and (iii) increase or decrease the bid depending on the demand-to-supply ratio when a seller’s expected yield increases. Inclusion of additional sellers lowers the optimal bids across the seller network, but it may not be optimal to bid at all sellers. For the two-seller problem, whether to diversify is a decision easily made by computing the expected benefit of bidding at each seller. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0190 .

Standby redundancy allocation for series and parallel systems

. The allocation of redundant components to a system is an efficient technique to improve its lifetime. In this article, we study the problem of two standby redundancy allocations for series and parallel systems, consisting of n-component (n≥2), in the sense of various stochastic orderings. It is assumed that components and redundancies follow a general lifetime distribution. For the case of allocating two redundancies to a series (parallel) system, we show that allocating the relatively stronger redundancy to the weaker component and the weaker redundancy to the stronger components can result in a system lifetime that is longer (shorter) in terms of the usual stochastic order. Moreover, various results associated with the hazard rate and reversed hazard rate orders are established. The outcomes of various papers are strengthened and generalized through our results. In addition, some applications are provided to illustrate our findings.

Some results on the odds rate order along with its statistical testing

Recently, Lando et al. [J. Statist. Plann. Inference 221 (2022) 313–325] define the odds rate order and show that it is stronger than the usual stochastic order, but weaker than the hazard rate order. The purpose of this paper is to investigate the other properties of this new stochastic order. Some characterization and preservation results and its relationship with other well-known stochastic orders are given. In addition, two relative ordering based on the odds rate functions are also defined and their relation with the odds rate order is investigated. Applications in statistical theory of reliability and a statistical test for the odds rate order are also included.

Generalized conditional spacings and their stochastic properties

<p>This paper introduces the concept of generalized conditional spacings and establishes partial order relations between different generalized spacings. First, we derive the survival function of the generalized conditional spacings. Second, we construct the stochastic and hazard rate order relationships between different generalized conditional spacings and generalized normal conditional spacings, considering parent distributions that belong to the decreasing failure rate (DFR) and increasing likelihood rate (ILR) classes. Finally, for parent distributions within the DFR class, we obtain the dispersive order between different conditional spacings, along with an inequality for the variance. Additionally, we present illustrative examples involving Pareto and Gamma distributions.</p>

The Topp-Leone Odd Burr X-G Family of Distributions: Properties and Applications

This paper proposes a new generalized family of distributions called the Topp-Leone odd Burr X-G (TLOBX-G) distribution and its special model, Topp-Leone odd Burr X-Weibull (TLOBX-W) is studied in detail. Structural properties are derived, including the hazard rate function, quantile function, density expansion, moments, R'enyi entropy, and order statistics. The maximum likelihood technique is used to estimate the parameters of the new family of distributions and a simulation study was carried out to assess the accuracy and consistency of these estimators. Finally, the applicability, usefulness, and flexibility of TLOBX-W distribution are illustrated using two real-life datasets.

Ordering properties of parallel and series systems with a general lifetime family of distributions for independent components under random shocks

Often, reliability systems suffer shocks from external stress factors, stressing the system at random. These random shocks may have non ignorable effects on the reliability of the system. In this article, we provide sufficient (and necessary) conditions on components’ lifetimes and their survival probabilities from random shocks for comparing the lifetimes of two parallel systems with independent components by means of usual stochastic and reversed hazard rate orders, when the matrix of parameters changes to another matrix in a mathematical sense. We also discuss stochastic comparisons of series systems with independent components under random shocks in the sense of usual stochastic and hazard rate orders, by using the concept of vector majorization and related orders. The results developed in this article generalize some known results in the literature.

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.

Optimal Grouping of Dependent Components in Parallel-Series and Series-Parallel Systems with Independent Subsystems Equipped with Starting Devices

In this paper, we consider parallel-series and series-parallel systems comprising dependent components that are drawn from a heterogeneous population consisting of m different subpopulations, and each subsystem is equipped with a starter device. We also make the assumption that the components within each subpopulation are dependent, while the subsystems themselves are independent. The joint distribution of these subsystems is modeled using an Archimedean copula. Our research considers a general setting in which each subpopulation has a different Archimedean copula for its dependence. By adopting this general setup, we investigate the stochastic, hazard rate, and reversed hazard rate orders between these systems. Furthermore, we provide several numerical examples to demonstrate all the theoretical results established in this study. These results broaden the scope of the known results in the existing literature.

Characterizations of stochastic ordering for non-negative random variables

Characterizations of the usual stochastic order, hazard rate order, reversed hazard rate order and likelihood ratio order under weaker conditions than log-concavity were provided for non-negative random variables. Our results generalized the previous work in Yu (2009).

Redundancy mechanisms to systems with statistically dependent component and redundancy lifetimes

AbstractThis study deals with redundancy mechanisms of coherent systems with statistically dependent component and redundancy lifetimes. We derive the distortion function of coherent system with a redundancy mechanism at component level, and then we present the usual stochastic order and hazard rate order on redundant system lifetimes. The main results not only enrich the framework of research in this line but also supplement some corresponding results in recent references. Several examples are presented to illustrate the results as well.