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.
Detecting outliers is an integral part of data analysis that sheds light on points that do not conform with the rest of the data. Whereas in univariate data, outliers appear at the extremes of the ordered sample, in the multivariate case they may be defined in many ways and are not generally based on an assumed statistical model. We present here methods for detecting multivariate outliers based on various definitions and illustrate their features by applying them to two sets of data. No single approach can be recommended over others, since each one aims at detecting outliers of a particular kind.
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