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Biased estimation of a variance

Authors:	Milan Merkle

Institution:	(1) Faculty of Electronic Engineering, University of Belgrade, P.O. Box 35-54, 11120 Belgrade, Yugoslavia

Abstract:	Summary Let $S^2 (\alpha ) = \frac{1}{{n + \alpha }}\mathop \Sigma \limits_{{\rm I} = 1}^n \left( {X_i - \hat \mu } \right)^2$ , whereX _i are i.i.d. random variables with a finite variance σ² and $\hat \mu$ is the usual estimate of the mean ofX _i. We consider the problem of finding optimal α with respect to the minimization of the expected value of \|S ²(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. This research was supported by Science Fund of Serbia, grant number 04M03, through Mathematical Institute, Belgrade.

Keywords:	Estimation of a variance biased-estimation chi square distribution Pitman's nearness
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