Nonparametric Neyman-Scott problems: Telescoping product methods

In this paper, we consider some generalizations of the Neyman-Scott problem of estimating a common variance in the presence of infinitely many mean nuisance parameters. For example, suppose observations are independent and stratified such that they have common distribution of the form F(x-μk) within the kth stratum. If both F and the location parameters μk are unknown, then estimates for F, obtained by centring the samples in each stratum and constructing an estimate directly from the pooled centred samples, are generally inconsistent unless the stratum sizes go to infinity. Similarly, nuisance parameters can arise from an exponential tilt of a common unknown distribution. Analogous estimators for the distribution produced by tilting the data 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 estimation of F using telescoping products. Partially multiplicative functions are introduced as a tool for the construction of counterexamples to the consistent estimation of F.