APPLICATIONES MATHEMATICAE 29,2 (2002), pp. 127–134 Alicja Jokiel-Rokita (Wroclaw) MINIMAX PREDICTION UNDER RANDOM SAMPLE SIZE Abstract. A class of minimax predictors of random variables with multi- nomial or multivariate hypergeometric distribution is determined in the case when the sample size is assumed to be a random variable with an unknown distribution. It is also proved that the usual predictors, which are minimax when the sample size is fixed, are not minimax, but they remain admissible when the sample size is an ancillary statistic with unknown distribution. 1. Introduction. In this paper we derive the minimax predictor of a random variable Y on the basis of the observation of a random variable X, in the case when X and Y have the multinomial or multivariate hypergeomet- ric distribution with the same unknown parameter. We assume that the loss function is given by (2) below and the sample size N is an ancillary statistic (i.e. a statistic whose distribution does not depend on the unknown pa- rameter). Moreover, we assume that this distribution is unknown. A widely held notion about ancillary statistics is that their distribution should be irrelevant to statistical inference. Therefore the minimax predictor for the case of fixed sample size may seem to be the best candidate for the mini- max predictor when the sample size is random. However, we prove that it is not minimax in our case. The first example of this ancillarity paradox was given by Brown (1990). Further examples were presented by He (1990) and Amrhein (1995). There are a lot of practical situations which emphasize the importance of considering a random sample size. For instance, there are so-called non- response models that take into account that for certain units in the sample, it may not be clear to which stratum they belong. Non-response typically 2000 Mathematics Subject Classification : Primary 62F10. Key words and phrases : admissibility, ancillarity paradox, loss function, minimax pre- dictor, multinomial distribution, multivariate hypergeometric distribution, risk function. [127]