Convergence and optimization of functional estimates in statistical modelling in Sobolev’s Hilbert spaces*

Abstract

The paper deals with the study of convergence and optimization of unbiased functional estimates in statistical modelling. We have obtained the estimates, which are optimal in Sobolev’s Hilbert spaces, for calculating the integrals dependent on a parameter and for calculating the families of functionals of the solution to the integral equation of the second kind. The results have been obtained in terms of the new concept proposed by the author in order to compare the efficiency of the functional estimates in the Monte Carlo method. We use the notation: E is the sign of expectation, V is variance, D is a differentiation operation. We denote by ║f║Hthe norm of a function f(x) in Sobolev’s Hilbert space. If fω=f(x,ω) is a family of functions, then ||f(­ ·,ω )||H denotes the corresponding family of norms. © 1995, VSP