Far East Journal of Theoretical Statistics
Volume 12, Issue 1, Pages 69 - 88
(January 2004)
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MIXTURES, HYBRID MIXTURES, CANONICAL FORMS, AND MATUSITA’S DISTANCE
K. O. Bowman (U. S. A.) and L. R. Shenton (U. S. A.)
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Abstract: First of all we introduce an orthogonal system associated with the mixture probability function. Then for maximum likelihood estimators, Parseval expansions are introduced related to logarithmic derivatives of the mixture probability function. In this way approximants are set up for the maximum likelihood covariance determinant.
For a hybrid mixture of Poisson-binomial, three parameters are involved and the covariance determinant approximant turns out to be a quartic for a ratio r, this relating to the means of the two components in the mixture. We prove explicitly that the zeros of this quartic are strictly complex; a mathematical proof is needed since computational values do not state the nature of the zeros. A study of a hybrid mixture of Poisson-negative binomial, two components, shows that the covariance approximant is a quartic, again having strictly complex zeros. In addition we briefly consider mixtures of gamma components, firstly with scale varying with fixed shape, and secondly, shape varying with fixed scale. Another mixture mentioned consists of two lognormal densities.
Lastly it is well known that estimation procedures for mixtures are sensitive to sample size, because of the closeness of the individual components. From a statistical point of view this characteristic is almost self-evident. There is therefore interest in a distance concept of probability function differences dual to Matusita [Amer. Math. Soc. 26 (1955), 631-640] who also introduces the dual notion of affinity between two probability functions. |
Keywords and phrases: affinity, covariance determinant, covariance matrix, distance, maximum likelihood, orthogonal system, Parseval expansion, persymmetric determinant, quartic zeros. |
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