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13 - The Normal Distribution
from Part V - Advanced Topics
Jianxin Wu, Nanjing University, China
Book:

Essentials of Pattern Recognition

Published online:

08 December 2020

Print publication:

19 November 2020, pp 293-315
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Summary

The normal distribution is the most widely used continuous distribution, but many of its relevant properties are a little bit advanced for an undergraduate course. Hence, Part IV introduces some of these advanced topics. This chapter devotes itself to properties of normal distributions: single- and multivariate normal distributions, moment and canonical parameterizations, sum and product, geometry and the Mahalanobis distance, and conditional distributions. We also show that with these properties, some algorithms will become much easier to understand. We use parameter estimation and the Kalman filter as two such examples.

Expansions for Repeated Integrals of Productswith Applications to the Multivariate Normal
Christopher S. Withers, Saralees Nadarajah
Journal:

ESAIM: Probability and Statistics / Volume 15 / 2011

Published online by Cambridge University Press:

05 January 2012, pp. 340-357

Print publication:

2011
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We extend Leibniz' rule for repeated derivatives of a product to multivariate integrals of a product.As an application we obtain expansions for P(a < Y < b) for Y ~ Np(0,V) andfor repeated integrals of the density of Y.When V−1y > 0 in R3 the expansion for P(Y < y) reduces toone given by [H. Ruben J. Res. Nat. Bureau Stand. B 68 (1964) 3–11]. in terms of the moments of Np(0,V−1).This is shown to be a special case of an expansion in terms of the multivariate Hermite polynomials.These are given explicitly.

The asymptotic theory of concomitants of order statistics
H. A. David, J. Galambos
Journal:

Journal of Applied Probability / Volume 11 / Issue 4 / December 1974

Published online by Cambridge University Press:

14 July 2016, pp. 762-770

Print publication:

December 1974
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In a random sample of n pairs (Xr, Yr), r = 1, 2, …, n, drawn from a bivariate normal distribution, let Xr:n be the rth order statistic among the Xr and let Y[r:n] be the Y-variate paired with Xr:n. The Y[r:n], which we call concomitants of the order statistics, arise most naturally in selection procedures based on the Xr:n. It is shown that asymptotically the k quantities k fixed, are independent, identically distributed variates. In addition, putting Rt,n for the number of integers j for which , the asymptotic distribution and all moments of n–1Rt, n are determined for t such that t/n → λ with 0 < λ < 1.