Estimation Based on Autopsy Data from Stress-Strength Experiments

Abstract

If Y is a random variable representing the breaking strength of a given material and X is a random variable measuring the stress placed on that material, then the probability that the material will survive the stress to which it is subjected is simply P = P(X < Y). In most applications involving stress-strength testing, the variables X and Y are modeled as independent with cumulative distribution functions F and G, respectively. Some of the highlights of the literature on the estimation of P will be reviewed. Our focus then shifts to the problem of estimating stress and strength distributions from autopsy data, for example, from data on welded steel bars from a collapsed bridge. The general framework studied involves a random pair (Y, Z), where Y is a real-valued random variable associated with the strength of randomly selected material and, given Y = y, Z is a Bernoulli variable with probability p = p (y), interpreted as the conditional probability of surviving stress Y = y. We will explore the situation in which inference concerning the strength distribution G is of interest, but the random variable Y cannot be observed directly. In such situations, one would seek to draw inferences concerning G from the observed values of Z. In a specific modeling scenario, we treat both classical and Bayesian versions of the problem of estimating model parameters, when identifiable, and develop a Bayesian treatment of the estimation of G and F in the presence of nonidentifiability. Our main purpose is to call attention to an interesting class of estimation problems and to provide a detailed example in which the coherence and feasibility of our estimation approach is demonstrated.

Key Words:

• Bayes
• exponential
• identifiability
• maximum likelihood
• Pareto
• reliability
• tensile strength

Additional information

Notes on contributors

Francisco J. Samaniego

Francisco J. Samaniegois Professor of Statistics at the University of California, Davis. His research interests include reliability modeling and inference, statistical decision theory, Bayesian methods, sampling techniques and statistical applications in engineering and public health. He is a Fellow of the American Statistical Association, the Institute of Mathematical Statistics and the Royal Statistical Society, and is an elected Member of the International Statistical Institute. He served as Editor of the Theory and Methods Section of the Journal of the American Statistical Association in 2002–2005.