Reversed Hazard Rate Research Articles

On the reversed hazard rate

Abstract The reversed hazard rate defined as the ratio of the density to the distribution function had attracted the attention of researchers only relatively recently. Being in a certain sense a dual function to an ordinary hazard rate, it still bears some interesting features useful in reliability analysis. One of its most important properties is the connection with the mean waiting time studied in this paper. The application to ordering of random variables via the proportional reversed hazard rate model is also considered. Possible applications are discussed.

The Use of Archimedean Copulas to Model Portfolio Allocations

A copula is a means of generating an n‐variate distribution function from an arbitrary set of n univariate distributions. For the class of portfolio allocators that are risk averse, we use the copula approach to identify a large set of n‐variate asset return distributions such that the relative magnitudes of portfolio shares can be ordered according to the reversed hazard rate ordering of the n underlying univariate distributions. We also establish conditions under which first‐ and second‐degree dominating shifts in one of the n underlying univariate distributions increase allocation to that asset. Our findings exploit separability properties possessed by the Archimedean family of copulas.

SOME RESULTS ON REVERSED HAZARD RATE ORDERING

Recently, the reversed hazard rate (RHR) function, defined as the ratio of the density to the distribution function, has become a topic of interest having applications in actuarial sciences, forensic studies and similar other fields. Here we establish results with respect to RHR ordering between the exponentiated random variables. We also address the ordering results between component redundancy and system redundancy. Both the cases of matching spares and non-matching spares are discussed. In case of matching spares, a sufficient condition has been given for component redundancy to be superior to the system redundancy with respect to the reversed hazard rate ordering for any coherent system.

Monotone Markov processes with respect to the reversed hazard rate ordering: an application to reliability

We consider a repairable system with a finite state space which evolves in time according to a Markov process as long as it is working. We assume that this system is getting worse and worse while running: if the up-states are ranked according to their degree of increasing degradation, this is expressed by the fact that the Markov process is assumed to be monotone with respect to the reversed hazard rate and to have an upper triangular generator. We study this kind of process and apply the results to derive some properties of the stationary availability of the system. Namely, we show that, if the duration of the repair is independent of its completeness degree, then the more complete the repair, the higher the stationary availability, where the completeness degree of the repair is measured with the reversed hazard rate ordering.

Monotone Markov processes with respect to the reversed hazard rate ordering: an application to reliability

We consider a repairable system with a finite state space which evolves in time according to a Markov process as long as it is working. We assume that this system is getting worse and worse while running: if the up-states are ranked according to their degree of increasing degradation, this is expressed by the fact that the Markov process is assumed to be monotone with respect to the reversed hazard rate and to have an upper triangular generator. We study this kind of process and apply the results to derive some properties of the stationary availability of the system. Namely, we show that, if the duration of the repair is independent of its completeness degree, then the more complete the repair, the higher the stationary availability, where the completeness degree of the repair is measured with the reversed hazard rate ordering.

SOME RESULTS ON REVERSED HAZARD RATE

In view of the growing importance of reversed hazard rate (RHR) in reliability analysis and stochastic modeling, we have considered different implicative relationships with respect to the monotonic behavior of RHR. In that context, a few characterizing properties have also been presented based on expected inactivity time.

The hazard rate and the reversed hazard rate orders, with applications to order statistics

In this paper we first point out a simple observation that can be used successfully in order to translate results about the hazard rate order into results about the reversed hazard rate order. Using it, we derive some interesting new results which compare order statistics in the hazard and in the reversed hazard rate orders; as well as in the usual stochastic order. We also simplify proofs of some known results involving the reversed hazard rate order. Finally, a few further applications of the observation are given.

Some results on the residual life at random time

In this paper, we consider the residual life at random time, i.e.X Y =X−Y\X>Y, whereX andY are non-negative random variables. We establish a number of stochastic comparison properties forX Y under various assumptions ofX andY. Under the assumption thatY has decreasing reverse hazard rate (DRHR), we show that ifX is in any one of the classes IFR, DFR, DMRL or IMRL thenX Y is in the same class asX. We also obtain some useful bounds for the distribution and the moment ofX Y . Because the idle time in classicalGI/G/1 queuing system can be regarded as the residual life at random time, the results obtained in this paper have applications in the study of such system.

A NOTE ON COMPARISONS OF k-OUT-OF-n SYSTEMS WITH RESPECT TO THE HAZARD AND REVERSED HAZARD RATE ORDERS

Let τk|n denote the lifetime of a k-out-of-n system constructed by using n components with independent (not necessarily identically distributed) lifetimes. It is shown that τk|n is smaller than τk−1|n−1 in the hazard rate order for any k and that τk|n−1 is smaller than τk|n in the reversed hazard rate order for any k. We thus strengthen and complement some results in Boland et al. [2] and Block et al. [1].

Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains

A continuous-time Markov chain on the non-negative integers is called skip-free to the right (left) if only unit increments to the right (left) are permitted. If a Markov chain is skip-free both to the right and to the left, it is called a birth–death process. Karlin and McGregor (1959) showed that if a continuous-time Markov chain is monotone in the sense of likelihood ratio ordering then it must be an (extended) birth–death process. This paper proves that if an irreducible Markov chain in continuous time is monotone in the sense of hazard rate (reversed hazard rate) ordering then it must be skip-free to the right (left). A birth–death process is then characterized as a continuous-time Markov chain that is monotone in the sense of both hazard rate and reversed hazard rate orderings. As an application, the first-passage-time distributions of such Markov chains are also studied.

Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains

A continuous-time Markov chain on the non-negative integers is called skip-free to the right (left) if only unit increments to the right (left) are permitted. If a Markov chain is skip-free both to the right and to the left, it is called a birth–death process. Karlin and McGregor (1959) showed that if a continuous-time Markov chain is monotone in the sense of likelihood ratio ordering then it must be an (extended) birth–death process. This paper proves that if an irreducible Markov chain in continuous time is monotone in the sense of hazard rate (reversed hazard rate) ordering then it must be skip-free to the right (left). A birth–death process is then characterized as a continuous-time Markov chain that is monotone in the sense of both hazard rate and reversed hazard rate orderings. As an application, the first-passage-time distributions of such Markov chains are also studied.

Preservation of some partial orderings under the formation of coherent systems

The reversed (backward) hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their reversed hazard rate functions. In this paper, we have given some sufficient conditions under which the ordering between the components with respect to the reversed hazard rate is preserved under the formation of coherent systems. We have also shown that these sufficient conditions are satisfied by k-out-of- n systems. Both the cases when components are identically distributed and not necessarily identically distributed are discussed. Some results for likelihood ratio order are also obtained. The parallel (series) systems of not necessarily iid components have been characterized by means of a relationship between the reversed hazard rate (hazard rate) function of the system and the reversed hazard rate (hazard rate) functions of the components.

The Reversed Hazard Rate Function

In this paper we discuss some properties of the reversed hazard rate function. This function has been shown to be useful in the analysis of data in the presence of left censored observations. It is also natural in discussing lifetimes with reversed time scale. In fact, ordinary hazard rate functions are most useful for lifetimes, and reverse hazard rates are natural if the time scale is reversed. Mixing up these concepts can often, although not always, lead to anomalies. For example, one result gives that if the reversed hazard rate function is increasing, its interval of support must be (—∞, b) where b is finite. Consequently nonnegative random variables cannot have increasing reversed hazard rates. Because of this result some existing results in the literature on the reversed hazard rate ordering require modification.Reversed hazard rates are also important in the study of systems. Hazard rates have an affinity to series systems; reversed hazard rates seem more appropriate for studying parallel systems. Several results are given that demonstrate this. In studying systems, one problem is to relate derivatives of hazard rate functions and reversed hazard rate functions of systems to similar quantities for components. We give some results that address this. Finally, we carry out comparisons for k-out-of-n systems with respect to the reversed hazard rate ordering.

Stochastic comparisons of parallel systems of heterogeneous exponential components

Let X 1, …, X n be independent exponential random variables with X i having hazard rate λ i , i = 1, …, n. Let λ = ( λ 1, …, λ n ). Let Y 1, …, Y n be a random sample of size n from an exponential distribution with common hazard rate λ = Σ n i = 1 λ i n . The purpose of this paper is to study stochastic comparisons between the largest order statistics X n: n and Y n: n from these two samples. It is proved that the hazard rate of X n: n is smaller than that of Y n: n . This gives a convenient upper bound on the hazard rate of X n: n in terms of that of Y n: n . It is also proved that Y n: n is smaller than X n: n according to dispersive ordering. While it is known that the survival function of X n: n is Schur convex in λ, Boland, El-Neweihi and Proschan [ J. Appl. Prohab. 31 (1994) 180–192] have shown that for n > 2, the hazard rate of X n: n is not Schur concave. It is shown here that, however, the reversed hazard rate of X n: n is Schur convex in λ.

Scheduling Independent Tasks to Minimize the Makespan on Identical Machines

In this paper we consider scheduling n tasks on m parallel machines where the task processing times are i.i.d. random variables with a common distribution function F. Scheduling is done by an a priori assignment of tasks to machines. We show that if the distribution function F is a Pólya frequency function of order 2 (decreasing reverse hazard rate) then the assignment that attempts to place an equal number of tasks on each machine achieves the stochastically smallest makespan among all assignments. The condition embraces many important distributions, such as the gamma and truncated normal distributions. Assuming that the task processing times have a common density that is a Pólya frequency function of order 2 (increasing likelihood ratio), then we find that flatter schedules have stochastically smaller makespans in the sense of the “joint” likelihood ratio.

On the order of tandem queues

We consider the optimal order of servers in a tandem queueing system withm stages, an unlimited supply of customers in front of the first stage, and a service buffer of size 1 but no intermediate storage buffers between the first and second stages. Service times depend on the servers but not the customers, and the blocking mechanism at the first two stages is manufacturing blocking. Using a new characterization of reversed hazard rate order, we show that if the service times for two servers are comparable in the reversed hazard rate sense, then the departure process is stochastically earlier if the slower server is first and the faster server is second than if the reverse is true. This strengthens earlier results that considered individual departure times marginally. We show similar results for the last two stages and for other blocking mechanisms. We also show that although individual departure times for a system with servers in a given order are stochastically identical to those when the order of servers is reversed, this reversibility property does not hold for the entire departure process.

Characterization of optimal order of servers in a tandem queue with blocking

Consider a tandem queueing system with m stages with no intermediate storage space between stages j and j + 1, j=1,…, m −1. There is an unlimited supply of customers in front of the first stage and the output buffer storage for stage m has an unlimited capacity. For this system we consider the problem of allocating m servers, one to each of these m stages such that the customer departure process from the system is stochastically maximized. In this regard we have shown that if the service times of the servers are comparable in the reversed hazard rate (or the usual stochastic) ordering then there exists an optimal allocation where the server allocated to the first stage has a larger mean service time than that assigned to the second stage. These results complement the recent results of Huang and Weiss (1990) and Yamazaki, Sakasegawa and Shanthikumar (1989).