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Distribution of the correlation coefficient for the class of bivariate elliptical models
Author(s) -
Ali Mir M.,
Joarder Anwarul H.
Publication year - 1991
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315434
Subject(s) - bivariate analysis , mathematics , fisher transformation , correlation coefficient , correlation , class (philosophy) , correlation ratio , distribution (mathematics) , statistics , statistical physics , mathematical analysis , physics , geometry , computer science , artificial intelligence
We consider n pairs of random variables (X 11 ,X 21 ) ,(X 12 ,X 22 ),… (X 1n ,X 2n ) having a bivariate elliptically contoured density of the form\documentclass{article}\pagestyle{empty}\begin{document}$$ K(n)|\Lambda |^{ - n/2} g\left\{ {\sum\limits_1^n {({\bf x}_{1j} - \theta _1 ,\,{\bf x}_{2j} - \theta _2 )\Lambda ^{ - 1} ({\bf x}_{1j} - \theta _1 ,\,{\bf x}_{2j} - \theta _2 )^{\rm T} } } \right\}, $$\end{document}where θ 1 θ 2 are location parameters and Δ = ((λ ik )) is a 2 × 2 symmetric positive definite matrix of scale parameters. The exact distribution of the Pearson product‐moment correlation coefficient between X 1 and X 2 is obtained. The usual case when a sample of size n is drawn from a bivariate normal population is a special case of the abovementioned model.