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1965
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Prem Chandra Consul, « The exact distribution of certain likelihood criteria useful in multivariate analysis », Bulletins de l'Académie Royale de Belgique, ID : 10.3406/barb.1965.65263
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Résumé
In this paper the exact distribution ƒ(l), of a statistic 'I' (0 ≤ l ≤ 1), whose t'-th moment is of a particular form, has been obtained, both for an integral index and fractional index, in the form of an infinite series from which the significance values of 'l' can be obtained to any degreee of accuracy. Then it has been shown that the distributions, obtained by Wald and Brookner (1941) for the Λ-criterion and by Verma (1951) for the Lmvc and Lvc criteria, are only particular cases of our result. By a further extension of the same method, the exact distribution the L-criterion, given by Votaw (1948) and Roy (1951), for testing the bipolarity of a covariance matrix, has also been obtained in the form of an infinite series, separately for integral and fractional indices. The actual values of the coefficients have also been calculated for some particular values of p and q and it has been shown that the total integral of the series, obtained by taking a few terms only, rapidly approaches the theoretical value 1 as more terms are taken into account. The 5% and 1% probability levels for the L-criterion can be easily computed by using tables of incomplete Beta function and by taking three or four terms only.