Moment data can be analytically completed

Abstract When an unknown (target) probability density function (pdf) is to be reconstructed from its moments with the aid of a multiparametric (or semi-parametric) representation model, a large number of constraints (moment equations) is needed, usually larger than the unknown parameters. However, there is a strong belief, expressed by various authors, see Fasino D, Inglese G (Rendiconti dell' Istituto di Matematica dell' Universita di Trieste XXVIII (1996) 183], and French JB [In: Dalton BJ, et al., Theory and application of moments problems in many-fermion systems (1979) 1], that most of the information defining a compactly supported pdf is usually contained in its first few moments. In addition, numerical experimentation has shown [Probab Engng Mech 17 (2002) 273] that the variation of the reconstructed pdf when higher-order moments vary within the recursively defined moment bounds, is weak, becoming insignificant as the moment order N increases. Based on these observations, we propose a method for extending a set of given moments, in cases that they are insufficient for solving the Hausdorff moment problem using a multiparametric model. Additional moments, compatible with the given ones, are obtained by exploiting upper and lower moment bounds. Then, two sequences of ‘dual’ (upper and lower) approximant pdfs are constructed, along with an easily realizable criterion of mutual convergence. The numerical performance and consistency of the solution algorithm have been illustrated for unimodal and bimodal pdfs.

[1]  L. Shapley,et al.  Geometry of Moment Spaces , 1953 .

[2]  Thomas F. Coleman,et al.  A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on Some of the Variables , 1992, SIAM J. Optim..

[3]  A. Salam.  On vector Hankel determinants , 2000 .

[4]  Kazimierz Sobczyk,et al.  Maximum entropy principle and non-stationary distributions of stochastic systems , 1996 .

[5]  Philip E. Gill,et al.  Practical optimization , 1981 .

[6]  J. B. French.  Elementary Principles of Spectral Distributions , 1980 .

[7]  J. Shohat,et al.  The problem of moments , 1943 .

[8]  N. Akhiezer,et al.  The Classical Moment Problem. , 1968 .

[9]  G. Talenti.  Recovering a function from a finite number of moments , 1987 .

[10]  E. E. Tyrtyshnikov.  How bad are Hankel matrices? , 1994 .

[11]  D. Fasino.  Spectral properties of Hankel matrices and numerical solutions of finite moment problems , 1995 .

[12]  J. Trębicki,et al.  Maximum entropy principle in stochastic dynamics , 1990 .

[13]  Alexander V. Tikhonov,et al.  Ill-Posed Problems in Natural Sciences , 1989 .

[14]  Gerassimos A. Athanassoulis,et al.  The truncated Hausdorff moment problem solved by using kernel density functions , 2002 .

[15]  Holger Dette,et al.  The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis , 1997 .

[16]  C. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[17]  D. Fasino,et al.  Recovering a probabilty density from a finite number of moments and local a priori information , 1996 .