On the Z-Representation of a Statistic

Abstract

The usual way of presenting the departure of the standardized sample mean of independent and identically distributed random variables from a normal variable is to write its distribution as an expansion with the standard normal distribution function as its leading term and remainder terms of order $n^{ - 1/ 2} ,n^{ - 1} $ etc. An alternative representation, which reveals the departure by functions of the normal variable itself, is introduced. This representation is not unique when the dimensionality of the variables is not one. However, it is shown that different representations do not affect the distribution of functions of the sample mean.