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Stieltjes classes for moment-indeterminate probability distributions

Part of: Integral transforms, operational calculus Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Jordan Stoyanov
Jordan Stoyanov*
Affiliation:
School of Mathematics and Statistics, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, UK. Email address: [email protected]
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Abstract

Let F be a probability distribution function with density f. We assume that (a) F has finite moments of any integer positive order and (b) the classical problem of moments for F has a nonunique solution (F is M-indeterminate). Our goal is to describe a , where h is a ‘small' perturbation function. Such a class S consists of different distributions Fε (fε is the density of Fε) all sharing the same moments as those of F, thus illustrating the nonuniqueness of F, and of any Fε, in terms of the moments. Power transformations of distributions such as the normal, log-normal and exponential are considered and for them Stieltjes classes written explicitly. We define a characteristic of S called an index of dissimilarity and calculate its value in some cases. A new Stieltjes class involving a power of the normal distribution is presented. An open question about the inverse Gaussian distribution is formulated. Related topics are briefly discussed.

Keywords

Problem of momentsStieltjes class of distributionsindex of dissimilaritynormal distributionlog-normal distributionexponential distributioninverse Gaussian distribution

MSC classification

Primary: 60E05: Distributions 44A60: Moment problems
Type
Part 5. Properties of random variables
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 281 - 294
Copyright
Copyright © Applied Probability Trust 2004 

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