Moment information for probability distributions, without solving the moment problem, II: Main-mass, tails and shape approximation

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:229

Article

Gavriliadis, P. N.¹  Athanassoulis, G. A.¹ 

[1] Natl Tech Univ Athens, Dept Naval Architecture & Marine Engn, Athens 15773, Greece

关键词: Chebyshev-Stieltjes-Markov inequality;    Moments;    Distribution functions;    Tail;    Christoffel function;    Approximation;   

DOI  :  10.1016/j.cam.2008.10.011

来源: Elsevier

【 摘 要 】

How much information does a small number of moments carry about the unknown distribution function? Is it possible to explicitly obtain from these moments some useful information, e.g., about the support, the modality, the general shape, or the tails of a distribution, without going into a detailed numerical solution of the moment problem? In this, previous and subsequent papers, clear and easy to implement answers will be given to some questions of this type. First, the question of how to distinguish between the main-mass interval and the tail regions, in the case we know only a number of moments of the target distribution function, will be addressed. The answer to this question is based on a version of the Chebyshev-Stieltjes-Markov inequality, which provides us with upper and lower, moment-based, bounds for the target distribution. Then, exploiting existing asymptotic results in the main-mass region, an explicit, moment-based approximation of the target probability density function is provided. Although the latter cannot be considered, in general, as a satisfactory solution, it can always serve as an initial approximation in any iterative scheme for the numerical solution of the moment problem. Numerical results illustrating all the theoretical statements are also presented. (C) 2008 Elsevier B.V. All rights reserved.

【 授权许可】

Free