The Hazard Potential

Abstract

The author thanks Hakon Gjessing for his help regarding the distribution of killing times of an integrated geometric Brownian motion process. The detailed comments by two referees are gratefully acknowledged. This is an expository article directed at reliability theorists, survival analysts, and others interested in looking at life history and event data. Here we introduce the notion of a hazard potential as an unknown resource that an item is endowed with at inception. The item fails when this resource becomes depleted. The cumulative hazard is a proxy for the amount of resource consumed, and the hazard function is a proxy for the rate at which this resource is consumed. With this conceptualization of the failure process, we are able to characterize accelerated, decelerated, and normal tests and are also able to provide a perspective on the cause of interdependent lifetimes. Specifically, we show that dependent life lengths are the result of dependent hazard potentials. Consequently, we are able to generate new families of multivariate life distributions using dependent hazard potentials as a seed. For an item that operates in a dynamic environment, we argue that its lifetime is the killing time of a continuously increasing stochastic process by a random barrier, and this barrier is the item's hazard potential. The killing time perspective enables us to see competing risks from a process standpoint and to propose a framework for the joint modeling of degradation or cumulative damage and its markers. The notion of the hazard potential generalizes to the multivariate case. This generalization enables us to replace a collection of dependent random variables by a collection of independent exponentially distributed random variables, each having a different time scale.

The Development of Parameter Estimation on Hazard Rate of Trivariate Weibull Distribution

“…this equation has been noted by (Singpurwalla, 2006). Unlike the univariate setup, there are various definitions of multivariate hazard (failure) rate functions.…”

An accumulator model for responses and response times in tests based on the proportional hazards model

“…The proportional hazards model can alternatively be derived from the assumption of a constant increase of knowledge log½K g ðtjhÞ ¼ log½K à B g ðtÞ þ b à 1g h and a randomly fluctuating response border C g . This derivation was proposed by Singpurwalla (2006). The proportional hazards model also follows from the assumption of an evidence counter model based on an inhomogeneous Poisson process (Otter, Allenby, & van Zandt, 2008).…”

“…The model is based on the assumption of two accumulators, the first reflecting the increase of knowledge over time and the second reflecting the tendency to discontinue working. With assumptions similar to Singpurwalla (2006), one can show that the model amounts to a proportional hazards model with competing risks. This is advantageous as such models have been used successfully in survival analysis for a long time.…”

Modeling and Simulation of Count Data

“…Still applied to epilepsy, results were presented where the last observation information was dichotomized depending on whether the seizure was present or absent. 24,44 Nonconstant event rate within time intervals As expressed before, the concept of hazard 45 is key in this context, since this rate determines when the events, ultimately counted, occur. Applying a model from the Poisson family assumes that this rate is constant within the intervals where the counting is made.…”