A note on the normalised moments of distributions with non-monotonic hazard rate

1981 ◽  
Vol 18 (2) ◽  
pp. 530-535
Author(s):  
H. L. MacGillivray
Keyword(s):  
Hazard Rate ◽  
Random Variables ◽  
Hazard Rates ◽  
Regular Property

For distributions of non-negative random variables with a monotonic hazard rate, it is well known that the normalised moments have the same sign-regular property as the distribution. This note extends this correspondence in properties for some common distributions with non-monotonic hazard rates, linking a change in sign-regular properties.

A note on the normalised moments of distributions with non-monotonic hazard rate

1981 ◽  
Vol 18 (02) ◽  
pp. 530-535 ◽  
Author(s):  
H. L. MacGillivray
Keyword(s):  
Hazard Rate ◽  
Random Variables ◽  
Hazard Rates ◽  
Regular Property

For distributions of non-negative random variables with a monotonic hazard rate, it is well known that the normalised moments have the same sign-regular property as the distribution. This note extends this correspondence in properties for some common distributions with non-monotonic hazard rates, linking a change in sign-regular properties.

Scheduling jobs by stochastic processing requirements on parallel machines to minimize makespan or flowtime

1982 ◽  
Vol 19 (1) ◽  
pp. 167-182 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Probability Distribution ◽  
Hazard Rate ◽  
Parallel Machines ◽  
Random Variables ◽  
Expected Value ◽  
Hazard Rates ◽  
Stochastic Processing ◽  
Identical Machines ◽  
Exponential Random Variables ◽  
Monotone Hazard Rate

A number of identical machines operating in parallel are to be used to complete the processing of a collection of jobs so as to minimize either the jobs' makespan or flowtime. The total processing required to complete each job has the same probability distribution, but some jobs may have received differing amounts of processing prior to the start. When the distribution has a monotone hazard rate the expected value of the makespan (flowtime) is minimized by a strategy which always processes those jobs with the least (greatest) hazard rates. When the distribution has a density whose logarithm is concave or convex these strategies minimize the makespan and flowtime in distribution. These results are also true when the processing requirements are distributed as exponential random variables with different parameters.

STOCHASTIC ORDER OF SAMPLE RANGE FROM HETEROGENEOUS EXPONENTIAL RANDOM VARIABLES

2008 ◽  
Vol 23 (1) ◽  
pp. 17-29 ◽  
Author(s):  
Peng Zhao ◽  
Xiaohu Li
Keyword(s):  
Hazard Rate ◽  
Stochastic Order ◽  
Random Variables ◽  
Hazard Rates ◽  
Minimum Order ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Usual Stochastic Order ◽  
Exponential Random Variables ◽  
Sample Range

Let X1, …, Xn be independent exponential random variables with their respective hazard rates λ1, …, λn, and let Y1, …, Yn be independent exponential random variables with common hazard rate λ. Denote by Xn:n, Yn:n and X1:n, Y1:n the corresponding maximum and minimum order statistics. Xn:n−X1:n is proved to be larger than Yn:n−Y1:n according to the usual stochastic order if and only if $\lambda \geq \left({\bar{\lambda}}^{-1}\prod\nolimits^{n}_{i=1}\lambda_{i}\right)^{{1}/{(n-1)}}$ with $\bar{\lambda}=\sum\nolimits^{n}_{i=1}\lambda_{i}/n$. Further, this usual stochastic order is strengthened to the hazard rate order for n=2. However, a counterexample reveals that this can be strengthened neither to the hazard rate order nor to the reversed hazard rate order in the general case. The main result substantially improves those related ones obtained in Kochar and Rojo and Khaledi and Kochar.

Scheduling jobs by stochastic processing requirements on parallel machines to minimize makespan or flowtime

1982 ◽  
Vol 19 (01) ◽  
pp. 167-182 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Probability Distribution ◽  
Hazard Rate ◽  
Parallel Machines ◽  
Random Variables ◽  
Expected Value ◽  
Hazard Rates ◽  
Stochastic Processing ◽  
Identical Machines ◽  
Exponential Random Variables ◽  
Monotone Hazard Rate

A number of identical machines operating in parallel are to be used to complete the processing of a collection of jobs so as to minimize either the jobs' makespan or flowtime. The total processing required to complete each job has the same probability distribution, but some jobs may have received differing amounts of processing prior to the start. When the distribution has a monotone hazard rate the expected value of the makespan (flowtime) is minimized by a strategy which always processes those jobs with the least (greatest) hazard rates. When the distribution has a density whose logarithm is concave or convex these strategies minimize the makespan and flowtime in distribution. These results are also true when the processing requirements are distributed as exponential random variables with different parameters.

scholarly journals A micro-analysis of Irish firm deaths during the financial crisis (2006–2010)

2020 ◽  
Vol 39 (1) ◽  
pp. 1-16
Author(s):  
Bernadette Power ◽  
Geraldine Ryan ◽  
Justin Doran
Keyword(s):  
Financial Crisis ◽  
Firm Size ◽  
Hazard Rate ◽  
Labour Force ◽  
Considerable Effect ◽  
Hazard Rates ◽  
Economic Cycle ◽  
Established Firms ◽  
Micro Analysis ◽  
Industry Conditions

AbstractThis paper examines differences in the hazard rates of young, established and mature firms during the financial crisis, using microdata from more than 300,000 Irish firms. The findings confirm that firm size at the time of the crisis had the largest impact on the probability of exit. The liability of smallness was pronounced in mature cohorts. Industry conditions had a considerable effect on the hazard rate of young cohorts, as opposed to mature counterparts. Interestingly, agglomeration raised the hazard rates of younger cohorts only. By contrast, attributes of the labour force of the region largely influenced the hazard rates of more established firms. Firms founded before the crisis were significantly less likely to exit in the aftermath of the crisis, in comparison with firms founded just before or during the crisis, whereas more mature firms seem to be more sensitive to the economic cycle.

A Bayes Continuous Reliability Growth Model with Two Categories of Failures

1997 ◽  
Vol 04 (01) ◽  
pp. 1-15
Author(s):  
Daming Lin ◽  
W. K. Chiu
Keyword(s):  
Competing Risks ◽  
Growth Model ◽  
Prior Distribution ◽  
Hazard Rate ◽  
Failure Modes ◽  
Weighted Average ◽  
Hazard Rates ◽  
Development Phase ◽  
Reliability Growth ◽  
Time Invariant

A Bayesian continuous reliability growth model is presented. It is assumed that the development phase of a product consists of m stages. In each stage, the failure mechanism of the product follows a competing risks model with two specific failure modes: inherent and assignable-cause. The hazard rate for each mode is time-invariant within one stage. Under the assumption that modifications of the product improve its reliability, we assign a reasonable joint prior distribution for the hazard rates. Then Bayesian analysis is carried out using this prior distribution. It turns out that the posterior pdf of the hazard rates of interest is just a weighted average of pdf's which have the same form as the prior pdf. A numerical example is given for illustration.

COMMENTS ON THE SURVEY BY BALAKRISHNAN AND ZHAO

2013 ◽  
Vol 27 (4) ◽  
pp. 445-449 ◽  
Author(s):  
Moshe Shaked
Keyword(s):  
Order Statistics ◽  
Negative Binomial ◽  
Random Variables ◽  
Hazard Rates ◽  
Proportional Hazard ◽  
Stochastic Comparisons ◽  
Exponential Random Variables ◽  
Binomial Random Variables

N. Balakrishnan and Peng Zhao have prepared an outstanding survey of recent results that stochastically compare various order statistics and some ranges based on two collections of independent heterogeneous random variables. Their survey focuses on results for heterogeneous exponential random variables and their extensions to random variables with proportional hazard rates. In addition, some results that stochastically compare order statistics based on heterogeneous gamma, Weibull, geometric, and negative binomial random variables are also given. In particular, the authors of have listed some stochastic comparisons that are based on one heterogeneous collection of random variables, and one homogeneous collection of random variables. Personally, I find these types of comparisons to be quite fascinating. Balakrishnan and Zhao have done a thorough job of listing all the known results of this kind.

scholarly journals Stochastic Order Relations Among Parallel Systems from Weibull Distributions

2015 ◽  
Vol 52 (01) ◽  
pp. 102-116 ◽  
Author(s):  
Nuria Torrado ◽  
Subhash C. Kochar
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Random Variables ◽  
Parallel Systems ◽  
Parallel System ◽  
Weibull Distributions ◽  
Scale Parameters ◽  
Order Relations ◽  
Stochastic Order Relations

Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.

A stochastic process for modeling failures of a system having a non-monotonic hazard rate function

2019 ◽  
Vol 233 (5) ◽  
pp. 731-746
Author(s):  
Lucianne Varn ◽  
Stefanka Chukova ◽  
Richard Arnold
Keyword(s):  
Stochastic Process ◽  
Hazard Rate ◽  
Repair Process ◽  
Rate Function ◽  
Hazard Rates ◽  
Failure Rates ◽  
Hazard Rate Function ◽  
Repairable Systems ◽  
System Failures ◽  
Quantities Of Interest

Reliability literature on modeling failures of repairable systems mostly deals with systems having monotonically increasing hazard/failure rates. When the hazard rate of a system is non-monotonic, models developed for monotonically increasing failure rates cannot be effectively applied without making assumptions on the types of repair performed following system failures. For instance, for systems having bathtub-shaped hazard rates, it is assumed that during the initial, decreasing hazard rate phase, all repairs are minimal. These assumptions on the type of general repair can be restrictive. In order to relax these assumptions, it has been suggested that general repairs in the initially decreasing phase can be modeled as “aging” the system. This approach however does not preserve the order of effectiveness of the types of general repair as defined in the literature. In this article, we develop a set of models to address these shortcomings. We propose a new stochastic process to model consecutive failures of repairable systems having non-monotonic, specifically bathtub-shaped, hazard rates, where the types of general repair are not restricted and the order of the effectiveness of the types of repair is preserved. The proposed models guarantee that a repaired system is at least as reliable as one that has not failed (or equivalently one that has been minimally repaired). To illustrate the models, we present multiple examples and simulate the failure-repair process and estimate the quantities of interest.

Geometric renewal convergence rates from hazard rates

2001 ◽  
Vol 38 (01) ◽  
pp. 180-194 ◽  
Author(s):  
Kenneth S. Berenhaut ◽  
Robert Lund
Keyword(s):  
Convergence Rate ◽  
Hazard Rate ◽  
Convergence Rates ◽  
Hazard Rates ◽  
Finite Support ◽  
Good Convergence ◽  
Renewal Sequence ◽  
Geometric Convergence ◽  
New Better Than Used ◽  
Better Than

This paper studies the geometric convergence rate of a discrete renewal sequence to its limit. A general convergence rate is first derived from the hazard rates of the renewal lifetimes. This result is used to extract a good convergence rate when the lifetimes are ordered in the sense of new better than used or increasing hazard rate. A bound for the best possible geometric convergence rate is derived for lifetimes having a finite support. Examples demonstrating the utility and sharpness of the results are presented. Several of the examples study convergence rates for Markov chains.

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