The Annals of Statistics

Log-Linear Models and Frequency Tables with Small Expected Cell Counts

Shelby J. Haberman

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In the case of frequency data, traditional discussions such as Rao (1973, pages 355-363, 391-412) consider asymptotic properties of maximum likelihood estimates and chi-square statistics under the assumption that all expected cell frequencies become large. If log-linear models are applied, these asymptotic properties may remain applicable if the sample size is large and the number of cells in the table is large, even if individual expected cell frequencies are small. Conditions are provided for asymptotic normality of linear functionals of maximum-likelihood estimates of log-mean vectors and for asymptotic chi-square distributions of Pearson and likelihood ratio chi-square statistics.

Article information

Ann. Statist., Volume 5, Number 6 (1977), 1148-1169.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62E20: Asymptotic distribution theory
Secondary: 62F05: Asymptotic properties of tests 62F10: Point estimation 62F25: Tolerance and confidence regions

Contingency tables log-linear models maximum likelihood chi-square tests asymptotic properties


Haberman, Shelby J. Log-Linear Models and Frequency Tables with Small Expected Cell Counts. Ann. Statist. 5 (1977), no. 6, 1148--1169. doi:10.1214/aos/1176344001.

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