Optimal Non-Parametric Prediction Intervals for Order Statistics with Random Sample Size

In many experiments, such as biology and quality control problems, sample size cannot always be considered as a constant value. Therefore, the problem of predicting future data when the sample size is an integer-valued random variable can be an important issue. This paper describes the prediction problem of future order statistics based on upper and lower records. Two different cases for the size of the future sample is considered as fixed and random cases. To do this, we first derive a general formula for the coverage probability of the prediction interval for each case. For the case that the sample size is a random variable, we consider two different distributions for the sample size, such as binomial and Poisson distributions and we study further details. The numerical computations are also given in this paper. Another purpose of this paper is to determine the optimal prediction interval for each case. Finally, the application of the proposed prediction interval is illustrated by analyzing the data in a real-world case study.