Distribution approximation and modelling via orthogonal polynomial sequences

Abstract

A general methodology is developed for approximating the distribution of a random variable on the basis of its exact moments. More specifically, a probability density function is approximated by the product of a suitable weight function and a linear combination of its associated orthogonal polynomials. A technique for generating a sequence of orthogonal polynomials from a given weight function is provided and the coefficients of the linear combination are explicitly expressed in terms of the moments of the target distribution. On applying this approach to several test statistics, we observed that the resulting percentiles are consistently in excellent agreement with the tabulated values. As well, it is explained that the same moment-matching technique can be utilized to produce density estimates on the basis of the sample moments obtained from a given set of observations. An example involving a well-known data set illustrates the density estimation methodology advocated herein.

Keywords:

• approximate distributions
• moment-matching techniques
• orthogonal polynomials
• percentiles
• test statistics
• data modelling
• density estimation

Acknowledgements

We would like to thank the three reviewers for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The financial support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. This research was also supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology [NRF-2013R1A1A2059794].