Estimation theory of a location parameter in small samples

Abstract

The topic of this thesis is estimation of a location parameter in small samples. Chapter 1 is an overview of the general theory of statistical estimates of parameters, with a special attention on the Fisher information, Pitman estimator and their polynomial versions. The new results are in Chapters 2 and 3 where the following inequality is proved for the variance of the Pitman estimator t_n from a sample of size n from a population F(x−\theta): nVar(t_n) >= (n+1)Var(t_{n+1}) for any n >= 1, only under the condition of finite second moments(even the absolute continuity of F is not assumed). The result is much stronger than the known Var(t_n) >= Var(t_{n+1}). Among other new results are (i) superadditivity of 1/Var(t_n) with respect to the sample size: 1/Var(t_{m+n}) >= 1/Var(t_m) + 1/Var(t_n), proved as a corollary of a more general result; (ii) superadditivity of Var(t_n) for a fixed n with respect to additive perturbations; (iii) monotonicity of Var(t_n) with respect to the scale parameter of an additive perturbation when the latter belongs to the class of self-decomposable random variables. The technically most difficult result is an inequality for Var(t_n), which is a stronger version of the classical Stam inequality for the Fisher information. As a corollary, an interesting property of the conditional expectation of the sample mean given the residuals is discovered. Some analytical problems arising in connection with the Pitman estimators are studied. Among them, a new version of the Cauchy type functional equation is solved. All results are extended to the case of polynomial Pitman estimators and to the case of multivariate parameters. In Chapter 4 we collect some open problems related to the theory of location parameters.