Bounds for the hazard rate and the reversed hazard rate of the convolution of dependent random lifetimes

2019 ◽  
Vol 56 (4) ◽  
pp. 1033-1043 ◽  
Author(s):  
Félix Belzunce ◽  
Carolina Martínez-Riquelme
Keyword(s):  
Recent Result ◽  
Upper Bound ◽  
Hazard Rate ◽  
Rate Function ◽  
Semiparametric Models ◽  
Hazard Rate Function ◽  
Reversed Hazard Rate ◽  
Parametric And Semiparametric Models ◽  
Reversed Hazard Rate Function

AbstractAn upper bound for the hazard rate function of a convolution of not necessarily independent random lifetimes is provided, which generalizes a recent result established for independent random lifetimes. Similar results are considered for the reversed hazard rate function. Applications to parametric and semiparametric models are also given.

The Reversed Hazard Rate Function

1998 ◽  
Vol 12 (1) ◽  
pp. 69-90 ◽  
Author(s):  
Henry W. Block ◽  
Thomas H. Savits ◽  
Harshinder Singh
Keyword(s):  
Time Scale ◽  
Hazard Rate ◽  
Rate Function ◽  
Hazard Rates ◽  
Hazard Rate Function ◽  
Reversed Hazard Rate ◽  
Rate Functions ◽  
Hazard Rate Ordering ◽  
Series Systems ◽  
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.

Reliability Concepts

2018 ◽  
pp. 1-22
Keyword(s):  
Hazard Rate ◽  
Failure Process ◽  
Reliability Function ◽  
Rate Function ◽  
Cumulative Distribution ◽  
Hazard Rate Function ◽  
Reliability Engineering ◽  
Probability Models ◽  
Engineering Design Process ◽  
Basic Concepts

The first chapter introduces basic concepts of Reliability and their relationships. Four probability functions—reliability function, cumulative distribution function, probability density function, and hazard rate function—that completely characterize the failure process are defined. Three failure rates—MTBF, MTTF, MTTR—that play important role in reliability engineering design process are explained here. The three patterns of failures, DFR, CFR, and IFR, are discussed with reference to the bathtub curve. Two probability models, Exponential and Weibull, are presented. Series and parallel systems and application areas of reliability are also presented.

ON THE QUASI-STATIONARY DISTRIBUTION OF SIS MODELS

2016 ◽  
Vol 30 (4) ◽  
pp. 622-639 ◽  
Author(s):  
Gaofeng Da ◽  
Maochao Xu ◽  
Shouhuai Xu
Keyword(s):  
Stationary Distribution ◽  
Upper Bound ◽  
Hazard Rate ◽  
State Of The Art ◽  
Upper Bounds ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Novel Method ◽  
Better Than ◽  
Quasi Stationary Distribution

In this paper, we propose a novel method for constructing upper bounds of the quasi-stationary distribution of SIS processes. Using this method, we obtain an upper bound that is better than the state-of-the-art upper bound. Moreover, we prove that the fixed point map Φ [7] actually preserves the equilibrium reversed hazard rate order under a certain condition. This allows us to further improve the upper bound. Some numerical results are presented to illustrate the results.

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