Biased estimation of a variance

Abstract

Let\(S^2 (\alpha ) = \frac{1}{{n + \alpha }}\mathop \Sigma \limits_{{\rm I} = 1}^n \left( {X_i - \hat \mu } \right)^2 \), whereXi are i.i.d. random variables with a finite variance σ2 and\(\hat \mu \) is the usual estimate of the mean ofXi. We consider the problem of finding optimal α with respect to the minimization of the expected value of |S2(σ)−σ2|k for variousk and with respect to Pitman's nearness criterion. For the Gaussian case analytical results are obtained and for some non-Gaussian cases we present Monte Carlo results regarding Pitman's criteron.