Statistical Inference Problems Based on the Moments of Distributions

The moments, if exist, are the simplest characteristics of any distribution. Thus it is natural to ask: Can we solve an inference problem in terms of the theoretical or empirical moments? In some cases the answer is yes,  in other cases it is not. We will discuss a variety of questions such as the role of the kurtosis and skewnes in characterizing a distribution, bounding a distribution based on the moments up to a fixed order, reconstruction of a distribution or its density based on the moments, comparison of the method of moments and the maximum likelihood method, etc.

It will become clear that in any inference problem the distribution involved must be uniquely determined by its moment sequence. This is one of the aspects of the classical moment problem. And this is why it is important to have convenient criteria allowing us to check weather or not a given distribution is unique in terms of the moments. Otherwise, to work with distributions which are non-unique by their moments is more than a risky business. A few recent and applicable results will be presented and discussed.