The large-sample distribution of the most fundamental of statistical summaries

We consider one of the most fundamental of statistical problems, namely that of inference for the mean, standard deviation and coefficients of skewness and kurtosis of an unknown univariate distribution. Assuming the distributional form of the parent population to be unknown, we focus our attention on moment-based inference. As is well-known, the method of moments estimates of the population measures under consideration are the sample mean, standard deviation and coefficients of skewness and kurtosis. Despite being some of the most frequently used of all statistical summaries, it comes as a surprise to find that their full joint distribution has not previously been studied in the literature. We derive a very general theoretical result for the large-sample asymptotic joint distribution of the four estimators and use simulation to explore the validity of the result as a means of approximating the biases, variances and covariances of the estimators for finite sample sizes. The theoretical result is then used to obtain asymptotically distribution-free inferential procedures for the population measures of original interest. Specifically, we propose and investigate the efficacy of bias-corrected and non-bias-corrected methods for point estimation and confidence set construction. We also discuss the relevance of the developed methodology both as an end in itself and as an aid to model formulation.