Small-sample comparisons for goodness-of-fit statistics in one-way multinomials with composite hypotheses

The small-sample behaviour of power-divergence goodness-of-fit statistics with composite hypotheses was evaluated with multinomial models of up to five cells and up to three parameters. Their performance was assessed by comparing asymptotic test sizes with exact test sizes obtained by enumeration in the near right tail, where 1-alpha is an element of (0.90, 0.95], and in the far right tail, where 1-alpha is an element of (0.95, 0.99]. The study addressed all combinations of power-diparse JAS312HH01.sgmvergence estimates of indices nu is an element of {-1/2, 0, 1/3, 1/2, 2/3, 1, 3/ 2} and power-divergence statistics of indices lambda is an element of {-1/2, 0, 1/3, 1/2, 2/3, 1, 3/2}. The results indicate that the asymptotic approximation is sufficiently accurate (by the criterion that the average exact size is no larger than +/-10% of the nominal asymptotic test size) in the near right tail when nu=0 and lambda=1/2, and in the far right tail when nu=0 and lambda=1/3, in both cases providing that the smallest expectation in the composite hypothesis exceeds 5. The only exception to this rule is the case of models that render a near-equiprobable composite hypothesis on the boundaries of the parameter space, where average exact sizes are usually quite different from nominal sizes despite the fact that the smallest expectation in these conditions is usually well above 5.