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Power Approximations to Multinomial Tests of Fit

F. C. Drost, W. C. M. Kallenberg, D. S. Moore and J. Oosterhoff
Journal of the American Statistical Association
Vol. 84, No. 405 (Mar., 1989), pp. 130-141
DOI: 10.2307/2289856
Stable URL: http://www.jstor.org/stable/2289856
Page Count: 12
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Power Approximations to Multinomial Tests of Fit
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Abstract

Multinomial tests for the fit of iid observations X1,...,Xn to a specified distribution F are based on the counts Ni of observations falling in k cells E1,...,Ek that partition the range of the Xj. The earliest such test is based on the Pearson (1900) chi-squared statistic: X2 = Σk i=1 (Ni - npi)2/npi, where pi = PF(Xj in Ei) are the cell probabilities under the null hypothesis. A common competing test is the likelihood ratio test based on LR = 2 Σk i=1 Nilog(Ni/npi). Cressie and Read (1984) introduced a class of multinomial goodness-of-fit statistics, Rλ, based on measures of the divergence between discrete distributions. This class includes both X2 (when λ = 1) and LR (when λ = 0). All of the Rλ have the same chi-squared limiting null distribution. The power of the commonly used members of the class is usually approximated from a noncentral chi-squared distribution that is also the same for all λ. We propose new approximations to the power that vary with the statistic chosen. Both the computation and results on asymptotic error rates suggest that the new approximations are greatly superior to the traditional power approximation for statistics Rλ other than the Pearson X2. The derivation of the limiting null distribution for the Cressie-Read statistics, following that for LR, is based on a Taylor series expansion of Rλ, in which X2 is the dominant term. The same expansion produces the traditional noncentral chi-squared power approximation by considering sequences of alternative distributions for the Xj that approach the hypotheses F at a suitable rate. Our power approximations are obtained from a Taylor series expansion that is valid for arbitrary sequences of alternatives. When linear and quadratic terms are retained, an accurate but computationally difficult approximation, Aλ, in terms of linear combinations of noncentral chi-squares is obtained. A second approximation, Bλ, in terms of a single noncentral chi-squared distribution results from averaging the coefficients in Aλ. This simple approximation performs well. In the important case of the statistic LR, Aλ = Bλ and this new noncentral chi-squared approximation is very accurate. Retaining only linear terms in the expansion produces an approximation Lλ based on a normal distribution; this is generally much inferior to Aλ and Bλ.

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