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  1. 2 Przegląd Statystyczny

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  1. 2 Westa M.

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  1. 2 2008

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1

ON USING THE GENERALIZED HELLWIG'S INEQUALITY FOR VERIFYING THAT A SYMMETRIC MATRIX IS A CORRELATION MATRIX

100%
Westa M.
Przegląd Statystyczny
|
2008
|
vol. 55
|
issue 2
64-77
EN
Hellwig (1976) proposed an inequality concerning the relationship between all pairwise correlation coefficients in the case of three variables. The generalised Hellwig's inequality (hereafter GHI) was derived by Borowiecki, Kaliszyk and Kolupa (1984). They argued that any symmetric k x k matrix, whichhas the foliowing properties: (1) k is greater than 3; (2) all elements on the main diagonal are units; (3) all elements outside the main diagonal are not greater than one in absolute value; is a correlation matrix if GHI is fulfilled for every element above the main diagonal (hereafter GHI criterion). These results were used by Dudek (2003). Methods of verification that a symmetric matrix with properties (1)-(3) is a correlation matrix (hereafter CM verification) were also considered by Hauke and Pomianowska (1987). They derived conditions (hereafter HP conditions) of using GHI in CM verification for a symmetric matrix of certain type. They did not consider the GHI criterion. In the present paper new conditions of using GHI in CM verification were derived. It was proved that (a) the GHI criterion properly indicates the correlation matrices only for k = 3; (b) if k is greater than 3 then the fulfilment of the GHI criterion is not a sufficient condition for a symmetric matrix with properties (1)-(3) to be a correlation matrix; (c) HP conditions are not true.
2

A NECESSARY AND SUFFICIENT CONDITION FOR A SYMMETRIC MATRIX TO BE A CORRELATION MATRIX

100%
Westa M.
Przegląd Statystyczny
|
2008
|
vol. 55
|
issue 3
33-46
EN
Borowiecki, Kolupa and Kaliszyk (1984) and Dudek (2003) proposed methods in which the generalised Hellwig's inequality is used for verifying that a symmetric matrix, which has the following properties: (1) - all elements on the main diagonal are units; (2) -all elements outside the main diagonal are not greater than one in absolute value, is a correlation matrix of certain variables. The authoress (see forthcoming paper) showed that this verification procedure may improperly indicate the correlation matrices. The theorems proved in the present paper define various forms of the necessary and sufficient condition for a symmetric matrix with properties (1)-(2) to be a correlation matrix. Among others things, it was shown that any symmetric 3x3 matrix with properties (1)-(2) is a correlation matrix if and only if its determinant is non-negative. Some results obtained generalize those given by Hauke and Pomianowska (1987) for correlation pair.
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