A comparison of the pearson-X2and log-likelihood-ratio statistics for small samples by means of probability ordering

The Pearson-X 2 and the log-likelihood-ratio (LLR) statistics are compared in the special case when they are used to test if the class probabilities of a multinomial distribution are equal. The exact rejection probabilities of the LLR statistic are examined to show that a multiplicative continuity correction is not appropriate. The possibility that different outcomes of a multinomial experiment can give rise to identical values of the Pearson-x 2 and/or log-likelihood-ratio is studied. To investigate differences between test-statistics under consideration, their ability to rank outcomes according to outcome-probabilities is compared with the ranking given by exact probabilities. It is seen that the LLR statistic is better in many respects and that its convergence to the x 2distribution is faster than that of the Pearson-X 2.