Abstract
The first hitting time of a boundary or threshold by the sample path of a stochastic process is the central concept of threshold regression models for survival data analysis. Regression functions for the process and threshold parameters in these models are multivariate combinations of explanatory variates. The stochastic process under investigation may be a univariate stochastic process or a multivariate stochastic process. The stochastic processes of interest to us in this report are those that possess stationary independent increments (i.e., Lévy processes) as well as the Esscher property. The Esscher transform is a transformation of probability density functions that has applications in actuarial science, financial engineering, and other fields. Lévy processes with this property are often encountered in practical applications. Frequently, these applications also involve a ‘cure rate’ fraction because some individuals are susceptible to failure and others not. Cure rates may arise endogenously from the model alone or exogenously from mixing of distinct statistical populations in the data set. We show, using both theoretical analysis and case demonstrations, that model estimates derived from typical survival data may not be able to distinguish between individuals in the cure rate fraction who are not susceptible to failure and those who may be susceptible to failure but escape the fate by chance. The ambiguity is aggravated by right censoring of survival times and by minor misspecifications of the model. Slightly incorrect specifications for regression functions or for the stochastic process can lead to problems with model identification and estimation. In this situation, additional guidance for estimating the fraction of non-susceptibles must come from subject matter expertise or from data types other than survival times, censored or otherwise. The identifiability issue is confronted directly in threshold regression but is also present when applying other kinds of models commonly used for survival data analysis. Other methods, however, usually do not provide a framework for recognizing or dealing with the issue and so the issue is often unintentionally ignored. The theoretical foundations of this work are set out, which presents new and somewhat surprising results for the first hitting time distributions of Lévy processes that have the Esscher property.
Highlights
Event times for systems and their components are of research interest in many fields including medicine, engineering, the natural sciences, economics and the social sciences.These events are important milestones, happenings or outcomes such as deaths, hospitalizations, divorces, business bankruptcies, engineering failures, and so on
Stats 2022, 5 the time to occurrence of the event. These failure times are often determined by an underlying stochastic process reaching a threshold, boundary or critical condition that triggers the failure event
To explain and study this phenomenon, we investigate a large collection of stochastic processes that are widely found in practical applications of threshold regression
Summary
Event times for systems and their components are of research interest in many fields including medicine, engineering, the natural sciences, economics and the social sciences. An individual in the study population may be at risk of eventual failure from the specific cause being investigated in the study We refer to this type of individual as a susceptible. The study population may include an individual who is not at risk of failure; an individual we call a non-suspectible Those individuals who are susceptible to failure may escape eventual failure. We present a setting for our investigation that involves a wide class of first hitting time models for study populations of mixed susceptibility. In our study we consider threshold regession applications that involve Lévy processes having the Esscher property (which is a common circumstance) In these contexts, the correct sign of the process drift parameter cannot be identified from censored failure data alone when the study population has some fraction of non-susceptibles. The problem usually goes unaddressed when other methods are used
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