Some uses of orthogonal polynomials in statistical inference

Abstract

Every random variable (rv) X (or random vector) with finite moments generates a set of orthogonal polynomials, which can be used to obtain properties related to the distribution of X. This technique has been used in statistical inference, mainly connected to the exponential family of distributions. In this paper a review of some of its more relevant uses is provided. The first one deals with properties of expansions in terms of orthogonal polynomials for the Uniformly Minimum Variance Unbiased Estimator of a given parametric function, when sampling from a distribution in the Natural Exponential Family of distributions with Quadratic Variance Function. The second one compares two relevant methods, based on expansions in Laguerre polynomials, existing in the literature to approximate the distribution of linear combinations of independent chi‐square variables.

Reliable Recurrence Algorithm for High-Order Krawtchouk Polynomials

“…In addition, 3D plots of the KP coefficients are given in this subsection. Flag=True; p ← p − 1 4: end if 5: x 0 ← N p, x 1 ← x 0 + 1 6: Compute K p 0 (x 0 ) using (10) 7: Compute K p 0 (x 1 ) using (12) 8: Compute K p 1 (x 0 ) and K p 1 (x 1 ) using ( 16) and ( 17) 9:…”

“…Similarly to image processing, speech signals processing is also essential [7], and involves several stages such as transfer [8], acquisition [9], and coding [10]. Pattern recognition, which is considered an automated process, is widely used in various applications such as computer vision [11], statistical data analysis [12], information retrieval [13], shot boundary detection [14], and bio-informatics [15]. However, the accuracy of extracting the significant features in these essential signal processing approaches is crucial [16].…”