SummaryThe principle of maximum likelihood is applied to the joint prediction and estimation of a future random variable and an unknown parameter. We assume dependence between present and future, and the approach is non-Bayesian. Our principal application is to the prediction of higher order statistics from lower ones in Type II censored random samples. Some simple criteria for existence and uniqueness of the predictor are given for this situation and the methods are illustrated with several examples.

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Summary An easily programmed recursive formula for the evaluation of the distribution function of ratios of linear combinations of independent exponential random variables is developed. This formula is shown to yield the probability that one team beats another in a contest we call the special gladiator game. This game generates tournaments which exhibit nontransitive dominance and have some surprising consequences. Similar results are obtained for a recursive formula based on the geometric distribution.