A Monte Carlo Method to Approximate Conditional Expectations based on a Theorem of Besicovitch: Application to Equivariant Estimation of the Parameters of the General Half-Normal Distribution

TL;DR

This paper introduces a Monte Carlo method based on Besicovitch's theorem for approximating conditional expectations, useful when densities are hard to compute, and applies it to estimate parameters of the half-normal distribution.

Contribution

It develops a new Monte Carlo approach for conditional expectations inspired by Besicovitch's theorem and applies it to derive novel estimators for the half-normal distribution parameters.

Findings

The method effectively approximates conditional expectations without requiring explicit densities.

New explicit forms for the minimum risk equivariant estimators of location and scale are provided.

Simulation results compare the new estimators with MLE and unbiased estimators, showing competitive performance.

Abstract

A natural Monte Carlo method to approximate conditional expectations in a probabilistic framework is justified by a general result inspired on the Besicovitch covering theorem on differentiation of measures. The method is specially useful when densities are not available or are not easy to compute. The method is illustrated by means of some examples and can also be used in a statistical setting to approximate the conditional expectation given a sufficient statistic, for instance. In fact, it is applied to evaluate the minimum risk equivariant estimator (MRE) of the location parameter of a general half-normal distribution since this estimator is described in terms of a conditional expectation for known values of the location and scale parameters. For the sake of completeness, an explicit expression of the the minimum risk equivariant estimator of the scale parameter is given. For all we know, these estimators have not been given before in the literature. Simulation studies are realized to compare the behavior of these estimators with that of maximum likelihood and unbiased estimators.