Inference and Prediction for a General Order Statistic Model with Unknown Population Size

Abstract

Suppose that the first n order statistics from a random sample of N positive random variables are observed, where N is unknown. This, the general order statistic model, has been applied to the study of market penetration, capture—recapture, burn-in in repairable systems, software reliability growth, the estimation of the number of individuals exposed to radiation, and the estimation of the number of unseen species. Inference is to be made about the unknown parameters, especially N, and future observations are to be predicted. A Bayes empirical Bayes approach to inference is presented. This permits the comparison of competing, perhaps nonnested, models for the distribution of the random variables in a natural way. It also provides easily implemented inference and prediction procedures that avoid the difficulties of non-Bayesian methods. One such difficulty is that the maximum likelihood estimator of N may be infinite. Results are given for the case in which vague prior information about the model parameters is approximated by limiting, improper, prior forms. Applications to three software reliability data sets indicate that the much-used exponential order statistic may give rather optimistic estimates of system reliability and that the not previously considered Weibull order statistic model seems promising for such applications.

Key Words:

• Bayes empirical Bayes
• Bayes factor
• Nonnested models
• Pareto order statistic model
• Software reliability
• Weibull order statistic model