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Simultaneous Estimation of Multinomial Cell Probabilities

Stephen E. Fienberg and Paul W. Holland
Journal of the American Statistical Association
Vol. 68, No. 343 (Sep., 1973), pp. 683-691
DOI: 10.2307/2284799
Stable URL: http://www.jstor.org/stable/2284799
Page Count: 9
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Simultaneous Estimation of Multinomial Cell Probabilities
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Abstract

A new estimator, p*, of the multinomial parameter vector is proposed, and it is shown to be a better choice in most situations than the usual estimator, $\mathbf{\hat p}$ (the vector of observed proportions). The risk functions (expected squared-error loss) of these two estimators are examined in three ways using: (a) exact calculations, (b) standard asymptotic theory, and (c) a novel asymptotic framework in which the number of cells is large and the number of observations per cell is moderate. The general superiority of p* over $\mathbf{\hat p}$ in large sparse multinomials is thus revealed. The novel asymptotic framework may also provide insight in other multinomial problems.

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