NONPARAMETRIC NEYMAN-SCOTT PROBLEMS - TELESCOPING PRODUCT METHODS

In this paper, we consider some generalizations of the Neyman-Scott pr oblem of estimating a common variance in the presence of infinitely ma ny mean nuisance parameters. For example, suppose observations are ind ependent and stratified such that they have common distribution of the form F(x - mu(kappa)) within the kappa th stratum. If both F and the location parameters mu(kappa) are unknown, then estimates for F, obtai ned by centring the samples in each stratum and constructing an estima te directly from the pooled centred samples, are generally inconsisten t unless the stratum sizes go to infinity. Similarly, nuisance paramet ers can arise from an exponential tilt of a common unknown distributio n. Analogous estimators for the distribution produced by tilting the d ata and pooling the strata will be inconsistent for similar reasons to the location parameter problem. In this paper, we concentrate on the case where the stratum sizes are fixed, and propose a method for the e stimation of F using telescoping products. Partially multiplicative fu nctions are introduced as a tool for the construction of counterexampl es to the consistent estimation of F.