Closer asymptotic approximations for the distributions of the power divergence goodness-of-fit statistics

Abstract

The members of the power divergence family of statistics\(2\left( {\lambda \left( {\lambda + 1} \right)} \right)^{ - 1} \sum\limits_{j = 1}^k {X_j \left[ {\left( {{{X_j } \mathord{\left/ {\vphantom {{X_j } {n\pi _{0j} }}} \right. \kern-\nulldelimiterspace} {n\pi _{0j} }}} \right)^2 - 1} \right]} \) all have an asymptotically equivalent χ2 distribution (Cressie and Read [1]). An asymptotic expansion for the distribution function is derived which shows that the speed of convergence to this asymptotic limit is dependent on λ. Known results for Pearson'sX 2 statistic and the log-likelihood ratio statistic then appear as special cases in a continuum rather than as separate (unrelated) expansions.