Optimal random sample size based on Bayesian prediction of exponential lifetime and application to real data

Choosing the sample size is a problem faced by anyone doing a survey of any type. “What sample size do we need?” is one of the most frequently asked questions of statisticians. The answer always starts with “It depends on...”. In this paper, we respond to this question by considering two criteria, total cost of experiment and mean squared prediction error in prediction problem. Towards this end, we discuss the problem of Bayesian predicting future observations from an exponential distribution based on an observed sample, when the information sample size is fixed as well as a random variable. Some distributions for the information sample size are considered and then for each case we find the parameter of distribution of the information sample size, such that the point predictor of a future order statistic has minimum mean squared prediction error when the total cost of experiment is bounded. To show the usefulness of our results, we present a simulation study. Finally, we apply our results to some real data sets in life testing.