Reversed Hazard Rate Order Research Articles

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 .

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).

Laplace transform ordering of bivariate inactivity times

<abstract><p>In this paper we consider the Laplace transform of the bivariate inactivity time. We show that a weak bivariate reversed hazard rate order is characterized by the Laplace transform of the bivariate inactivity times in two different frames. The results are used to characterize the weak bivariate reversed hazard rate order using the weak bivariate mean inactivity time order. The results are also used to characterize the decreasing bivariate reversed hazard rate property using the Laplace transform of the bivariate inactivity time.</p></abstract>

Ordering Results for Order Statistics from Heterogeneous Log-Logistic Distributions

SYNOPTIC ABSTRACT The largest and the smallest order statistics respectively represent the lifetime of a parallel and a series system. Various stochastic orders such as the usual stochastic order, hazard rate order and the reversed hazard rate order are used to obtain a stochastically better system. In the present communication, we consider stochastic comparison of the largest and the smallest order statistics arising from heterogeneous log-logistic distributions. First, we treat the case when the components do not receive random shocks. In other case, we assume that the components receive random shocks. The comparisons are studied in terms of the dispersive, usual stochastic, hazard rate and the reversed hazard rate orders. Majorization-based sufficient conditions are obtained to compare the order statistics. In addition, to illustrate the results, several numerical examples are presented.

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.

Stochastic comparison on load‐sharing systems with one statistically dependent redundancy

AbstractIn this article, we study a load‐sharing redundancy system in the presence of statistical dependence between base and redundancy component lifetimes when they are under partial workload. We derive the system reliability function in terms of reliability functions and the copula function of the base and redundancy component lifetimes. Further, in the setting of some aging properties, the base component and absolute continuous copula, we develop the hazard rate order, the reversed hazard rate order, and the likelihood ratio order between the system lifetime and the base component lifetimes. Two typical scenarios with independence between the base and redundancy component lifetimes are discussed in particular. Several numerical examples are presented to illustrate the theoretical findings as well.

Stochastic comparisons of parallel systems with heterogeneous exponentiated half logistic-F components

This study endeavors to compare two parallel systems delineated via exponentiated half logistic-F distributions with both identical (F) and non-identical (F 1 and F 2) baseline distribution functions. The comparisons are carried out with respect to the usual stochastic order as well as the reversed hazard rate order through the majorized shape parameters of the distributions.

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.

ORDERINGS OF FINITE MIXTURE MODELS WITH LOCATION-SCALE DISTRIBUTED COMPONENTS

In this paper, we consider finite mixture models with components having distributions from the location-scale family. We then discuss the usual stochastic order and the reversed hazard rate order of such finite mixture models under some majorization conditions on location, scale and mixing probabilities as model parameters.

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.

Component level versus system level at active redundancies for coherent systems with dependent heterogeneous components

This article investigates the issue of stochastic comparison of multi-active redundancies at component level versus system level. In the case of matching and non matching spares, we present some interesting comparison results of coherent systems with dependent heterogeneous components in the sense of the usual stochastic ordering, the hazard rate ordering, and the reversed hazard rate ordering. Several numerical examples are provided to illustrate the main results established here.

Application of weighted and unordered majorization orders in comparisons of parallel systems with exponentiated generalized gamma components

Consider two parallel systems, say $A$ and $B$, with respective lifetimes $T_{1}$ and $T_{2}$ wherein independent component lifetimes of each system follow exponentiated generalized gamma distribution with possibly different exponential shape and scale parameters. We show here that $T_{2}$ is smaller than $T_{1}$ with respect to the usual stochastic order (reversed hazard rate order) if the vector of logarithm (the main vector) of scale parameters of System $B$ is weakly weighted majorized by that of System $A$, and if the vector of exponential shape parameters of System $A$ is unordered mojorized by that of System $B$. By means of some examples, we show that the above results can not be extended to the hazard rate and likelihood ratio orders. However, when the scale parameters of each system divide into two homogeneous groups, we verify that the usual stochastic and reversed hazard rate orders can be extended, respectively, to the hazard rate and likelihood ratio orders. The established results complete and strengthen some of the known results in the literature.

Stochastic comparisons of series and parallel systems with dependent and heterogeneous Topp-Leone generated components

<abstract> <p>In this paper, we carry out stochastic comparisons of lifetimes of series and parallel systems with dependent heterogeneous Topp-Leone generated components. The usual stochastic order and the reversed hazard rate order are developed for the parallel systems with the help of vector majorization under Archimedean copula dependence, and the results for the usual stochastic order are also obtained for the series systems. Finally, some numerical examples are provided to illustrate the effectiveness of our theoretical findings.</p> </abstract>

Ordering extremes of exponentiated location-scale models with dependent and heterogeneous random samples

This paper is devoted to some ordering results for the largest and the smallest order statistics arising from dependent heterogeneous exponentiated location-scale random observations. We assume that the sets of observations are sharing a common or different Archimedean copula(s). Sufficient conditions for which the usual stochastic order and the reversed hazard rate order between the extreme order statistics hold are derived. Various numerical examples are provided for the illustration of the proposed results. Finally, some applications of the comparison results in engineering reliability and auction theory are presented.

Stochastic comparisons of series, parallel and [formula omitted]-out-of-[formula omitted] systems with heterogeneous bathtub failure rate type components

In this paper, we explore stochastic comparisons of order statistics arising from a two parameter distribution with bathtub shaped failure rate. We obtain detailed results in the contexts of the usual stochastic order, the hazard rate order, the reversed hazard rate order and the likelihood ratio order for series, parallel and k-out-of-n systems. Results involving multivariate chain majorization are also established.

Ordering properties of largest order statistics from independent and heterogeneous Dagum populations

The Dagum distribution has been extensively used to model income data, and its features have been appreciated in economics and financial studies. In this article, we discuss ordering properties of largest order statistics from independent and heterogeneous Dagum populations. We present some sufficient conditions for stochastic comparisons between largest order statistics in terms of the reversed hazard rate order, the usual stochastic order, the convex order, the likelihood ratio order and the dispersive order. Several numerical examples are presented to illustrate the results established here.

Hazard rate and reversed hazard rate orders on extremes of heterogeneous and dependent random variables

We develop sufficient conditions for the hazard rate order on minimums of sample with Archimedean survival copulas and having proportional hazard rates or scales, and the reversed hazard rate order on maximums of sample with Archimedean copulas and have proportional reversed hazard rates, respectively. Also, we present applications of the hazard rate order on sample minimums in engineering reliability and actuarial risk.

On Redundancy Allocation to Series and Parallel Systems of Two Components

For two components and one standby redundancy, we develop a characterization on the hazard rate order and the reversed hazard rate order of the redundant system lifetime in the context of mutually independent components lifetimes. Also, the likelihood ratio order is derived on the lifetime of the series system with two components lifetimes and two matched active redundancies lifetimes both following the proportional hazard model.

Ordering results of the smallest and largest order statistics from independent heterogeneous exponentiated gamma random variables

In this paper, we discuss stochastic comparisons of lifetimes of series and parallel systems with heterogeneous exponentiated gamma components. The results established here are developed in two directions. First, when a system possibly has different shape and scale parameters and the matrix of those different parameters following the chain majorization order, we study the reversed hazard rate order of parallel systems and the usual stochastic order of series systems. Next, by using the concept of vector majorization, we establish the usual stochastic order of series systems and the reversed hazard rate order of parallel systems.

On the optimal allocation of active redundancies in series system

ABSTRACTThis paper studies the allocations of two non identical active redundancies in series systems in terms of the reversed hazard rate order and hazard rate order, which generalizes some results built in Valdés and Zequeira (2003, 2006).