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

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.