Polynomial probability distribution estimation using the method of moments
Authors: Lars Mattsson
Joakim Munkhammar
Jesper Rydén
Journal: PLoS ONE (2017)
Discipline: Multidisciplinary
Keywords: Polynomial approximations; Probability density functions; Method of moments; Distribution families; Normal
Affiliations: BEESG, Department of Engineering Sciences/Uppsala University, Uppsala, Sweden., Nordita, KTH Royal Institute of Technology and Stockholm University, Stockholm, Sweden., Department of Mathematics, Uppsala University, Uppsala, Sweden.

Abstract:  We suggest a procedure for estimating Nth degree polynomial approximations to unknown (or known) probability density functions (PDFs) based on N statistical moments from each distribution. The procedure is based on the method of moments and is setup algorithmically to aid applicability and to ensure rigor in use. In order to show applicability, polynomial PDF approximations are obtained for the distribution families Normal, Log-Normal, Weibull as well as for a bimodal Weibull distribution and a data set of anonymized household electricity use. The results are compared with results for traditional PDF series expansion methods of Gram-Charlier type. It is concluded that this procedure is a comparatively simple procedure that could be used when traditional distribution families are not applicable or when polynomial expansions of probability distributions might be considered useful approximations. In particular this approach is practical for calculating convolutions of distributions, since such operations become integrals of polynomial expressions. Finally, in order to show an advanced applicability of the method, it is shown to be useful for approximating solutions to the Smoluchowski equation. 

Retracted: No

PubPeer comments:
 Although it is true that any continuous function on a closed interval may be approximated arbitrary closely by a polynomial, a single polynomial (necessarily of high order) is seldom used to approximate a probability density function. Splines, kernel methods, and some other methods serve the purpose and perform better. One reason is that the tail behavior of a probability density function is hard to be approximated by a single polynomial. This phenomenon is illustrated by the author themselves in Fig.2. Another reason is that the moment matching method used to determine the coefficients typically leads to a matrix with a large condition number and hence not numerically robust when the moment values on the right hand side are estimated from data. This issue may be observed from eq.(6). Besides, it is a common practice in statistics to avoid using high order moments for estimation. , 
 #1 The method does have weaknesses with tails, but works just fine for most applications. It is a comparatively new model, but that has already been successfully used in a series of papers, e,g, for modeling solar irradiance , fast adaptive filtering (doi: 10.1109 TIP.2018.2871597), stochastic bus injection . Also it was used for obtaining solutions to the Smoluchowski equation in the original paper. , 
 It is interesting to point out that this method was also applied in decision making in an unexpected manner. See https: pubpeer.com publications 2855F79B5FDCA85D55D4C0CA2B49B7 ,