The Hazard Potential

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.

[1]  E. Gumbel.  Bivariate Exponential Distributions , 1960 .

[2]  R. M. Blumenthal,et al.  Markov processes with identical hitting distributions , 1962 .

[3]  Bernard M. Levin,et al.  Statistical theory of reliability , 1963 .

[4]  L. Schmetterer.  Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete. , 1963 .

[5]  J. Kingman.  On doubly stochastic Poisson processes , 1964, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  I. Olkin,et al.  A Multivariate Exponential Distribution , 1967 .

[7]  P. Suppes.  A Probabilistic Theory Of Causality , 1970 .

[8]  David R. Cox,et al.  Regression models and life tables (with discussion , 1972 .

[9]  E. Çinlar.  Markov additive processes. II , 1972 .

[10]  T. Speed,et al.  A note on random times , 1973 .

[11]  A. W. Marshall,et al.  Shock Models and Wear Processes , 1973 .

[12]  Albert W. Marshall,et al.  Some comments on the hazard gradient , 1975 .

[13]  J. Grandell.  Doubly stochastic Poisson processes , 1976 .

[14]  Richard E. Barlow,et al.  Statistical Theory of Reliability and Life Testing: Probability Models , 1976 .

[15]  E. Çinlar.  On increasing continuous processes , 1979 .

[16]  Austin J. Lemoine,et al.  On failure modeling , 1985 .

[17]  D. Lindley,et al.  Multivariate distributions for the life lengths of components of a system sharing a common environment , 1986, Journal of Applied Probability.

[18]  Odd O. Aalen,et al.  Dynamic modelling and causality , 1987 .

[19]  Süleyman Özekici,et al.  Reliability of Complex Devices in Random Environments. , 1987 .

[20]  Elja Arjas,et al.  A note on random intensities and conditional survival functions , 1988, Journal of Applied Probability.

[21]  Michael L. Wenocur.  A reliability model based on the gamma process and its analytic theory , 1989, Advances in Applied Probability.

[22]  Kazimierz Sobczyk.  Stochastic models for fatigue damage of materials , 1989 .

[23]  O. Aalen.  A linear regression model for the analysis of life times. , 1989, Statistics in medicine.

[24]  P. Protter.  Stochastic integration and differential equations : a new approach , 1990 .

[25]  J. L. Bogdanoff,et al.  Application of Physical Laws to Parameter Estimation for Probabilistic Models of Cumulative Damage , 1990 .

[26]  P. Protter.  Stochastic integration and differential equations , 1990 .

[27]  K. A. Doksum,et al.  Degradation rate models for failure time and survival data , 1991 .

[28]  Y. Kebir.  On hazard rate processes , 1991 .

[29]  M. Yor.  On some exponential functionals of Brownian motion , 1992, Advances in Applied Probability.

[30]  Nozer D. Singpurwalla,et al.  Multivariate distributions induced by dynamic environments , 1993 .

[31]  Nozer D. Singpurwalla,et al.  Survival in Dynamic Environments , 1995 .

[32]  Simon P. Wilson,et al.  The exponentiation formula of reliability and survival: Does it always hold? , 1995, Lifetime data analysis.

[33]  Roger B. Nelsen,et al.  Copulas, Characterization, Correlation, and Counterexamples , 1995 .

[34]  S. D. Durham,et al.  Cumulative damage models for system failure with application to carbon fibers and composites , 1997 .

[35]  G. A. Whitmore,et al.  Failure Inference From a Marker Process Based on a Bivariate Wiener Model , 1998, Lifetime data analysis.

[36]  V. Bagdonavičius,et al.  Model Building in Accelerated Experiments , 1999 .

[37]  L. Pettit,et al.  Bayesian analysis for inverse gaussian lifetime data with measures of degradation , 1999 .

[38]  Georgia-Ann Klutke,et al.  Lifetime-characteristics and inspection-schemes for Levy degradation processes , 2000, IEEE Trans. Reliab..

[39]  Peng Huang,et al.  Stochastic Models in Reliability , 1999, Technometrics.

[40]  L. Gavrilov,et al.  The reliability theory of aging and longevity. , 2001, Journal of theoretical biology.

[41]  Nils Lid Hjort,et al.  Frailty models based on Lévy processes , 2003, Advances in Applied Probability.

[42]  Jeffrey S. Rosenthal,et al.  On the collapsibility of lifetime regression models , 2003, Advances in Applied Probability.

[43]  Nozer D. Singpurwalla,et al.  Specifying interdependence in networked systems , 2004, IEEE Transactions on Reliability.

[44]  D.,et al.  Regression Models and Life-Tables , 2022 .