Comparisons of Chisquare, Edgeworth Expansions and Bootstrap Approximations to the Distribution of the Frequency Chisquare

Accuracies of aggregated local Edgeworth expansions, nonparametric bootstrap, and the classical chisquare, as approximations of the distribution of Pearson's frequency chisquare statistic, are examined in this paper. Modern computing technology allows one to locate lattice points in an ellipsoid and aggregate expansions of point masses at these points, so that local expansions provide a feasible alternative. There are a number of surprises. Although the asymptotic theory gives a smaller order of error for aggregated local expansions than the chisquare, different phenomena show up in actual computations with small samples and extreme asymmetry. Notwithstanding the proverbial minimum cell count 5, the classical chisquare performs remarkably well for fairly small sample sizes often with an expected cell frequency lower than or equal to one, outdoing all its competitors in such cases. On the other hand, with moderate asymmetry and/or much larger sample sizes, local expansions yield better approximations. The chisquare approximation also seems to have an edge over the bootstrap, although the two are shown to have the same asymptotic order of error.