On Properties of the (Φ, a)-Power Divergence Family with Applications in Goodness of Fit Tests

摘要

In this paper we unify the different measures of divergence by introducing a general class of measures of divergence, the (Φ, a) −power divergence family and investigate its main properties including the limiting property, the order preserving property, and the quadratic convergence. For the practical implications of the proposed class of measures, we examine its use in goodness of fit tests for multinomial populations. In particular, a test statistic for goodness of fit tests based on the proposed family of measures is investigated for small sample sizes and various multinomial distributions that include symmetric, skewed and equiprobable models. The proposed statistic appears to work well in all cases considered as opposed to other traditional tests including the traditional chi-squared Pearson’s test, which may work well in some but not all situations.