Smoothed Versions of Statistical Functionals from a Finite Population

We will consider the central limit theorem for the smoothed version of statistical functionals in a finite population. For the infinite population, Reeds (1976) and Fernholz (1983) discuss the problem under the conditions of Hadamard differentiability of the statistical functionals and derive Taylor type expansions. Lindeberg-Feller's central limit theorem is applied to the leading term, and controlling the remainder terms, the central limit theorem for the statistical functionals are proved. We will modify Fernholz's method and apply it to the finite population with smoothed empirical distribution functions, and we will also obtain Taylor type expansions. We then apply the Erdös-Rényi central limit theorem to the leading linear term to obtain the central limit theorem. We will also obtain sufficient conditions for the central limit theorem, both for the smoothed influence function, and the original non-smoothed versions. Some Monte Carlo simulation results are also included.