The truncated Hausdorff moment problem solved by using kernel density functions

In this work, the problem of an efficient representation and its exploitation to the approximate determination of a compactly supported, continuous probability density function (pdf) from a finite number of its moments is addressed. The representation used is a finite superposition of kernel density functions. This representation preserves positivity and can approximate any continuous pdf as closely as it is required. The classical theory of the Hausdorff moment problem is reviewed in order to make clear how the theoretical results as, e.g. the moment bounds, can be exploited in the numerical procedure. Various difficulties arising from the well-known ill-posedness of the numerical moment problem have been identified and solved. The kernel coefficients of the pdf expansion are calculated by solving a constrained, non-negative least-square problem. The consistency, numerical convergence and robustness of the solution algorithm have been illustrated by numerical examples with unimodal and bimodal pdfs. Although this paper is restricted to univariate, compactly supported pdfs, the method can be extended to general pdfs either univariate or multivariate, with finite or infinite support.