Covarianza fra media campionaria e differenza media: aspetti formali e relazioni con l’asimmetria

Abstract

In a recent work Polisicchio and Zenga (1996) deduced a simple formula for the covariance betwen sample mean and (sample) mean difference in case of finite population. Studying the joint distribution of sample mean and (sample) mean difference derived from three simple discrete populations, the same authors observed a concordance between the sign of this covariance and the sense of asymmetry of the populations in case.
In this work simple and equivalent formulas of the covariance between sample mean and (sample) mean difference will be deduced in the continuos case. These formulas will then be applied to the uniform, exponential, Pareto and “inverse” Pareto random variables. These applications show the same relationship between the asymmetry and the sign of the covariance:
Then, it will be shown that monotonicity of point asymmetries is relevant to check the sign of the above mentioned covariance, both for continuos and discrete case.