The Pearson’s chi-square and the log-likelihood ratio chi-square statistics are fundamental tools in multinomial goodness-of-fit testing. Cressie and Read (1984) constructed a general family of divergences which includes both statistics as special cases. This family is indexed by a single real parameter. Divergences at one end of the scale are powerful against deviations of one type while being poor against deviations of the other type. The reverse property holds for divergences at the other end of the scale. Several other families of divergences available in the literature also show similar behavior. We present several inlier control techniques in the context of multinomial goodness-of-fit testing which generate procedures having reasonably high powers for both kinds of alternatives. We explain the motivation behind the construction of the inlier modified test statistics, establish the asymptotic null distribution of the inlier modified statistics and explore their performance through simulation and real data examples to substantiate the theory developed.
"Multinomial goodness-of-fit tests under inlier modification." Electron. J. Statist. 5 1846 - 1875, 2011. https://doi.org/10.1214/11-EJS658