A Note on an Estimator for the Variance That Utilizes the Kurtosis

Abstract

Abstract An estimator (S W 2) for the variance is developed by minimizing the mean squared error (MSE) using a generalized weight for the sum of squares instead of 1/(n − 1). The optimal divisor found is (n + 1) + (α4 − 3) (n 1 1)/n, where α4 is the kurtosis. For the normal distribution (α4 = 3), the divisor becomes n + 1. Generally, for kurtosis greater than 3 the divisor will be greater than n + 1 and for kurtosis less than 3 the divisor will be less than n + 1. Using n + 1 as a divisor will result in a smaller MSE for distributions with α4 > 3.