On the Characteristic Functions of the Distributions of Estimates of Various Deviations in Samples from a Normal Population

Abstract

An explicit formula for the characteristic function of the deviation $\frac{1}{n} \sum_n {k=1}\|X_k - \bar X\|^\alpha,\quad\alpha > 0,$ is derived for samples from a normal population. For $\alpha = 1$ one can calculate the probability density function but the result does not seem to be in complete agreement with a recent formula of Goodwin [1].