Two facets of the moran statistic

Abstract

The asymptotic distributions of statistics are often used to construct finite sample approximations for their distributions. However statistics which arise in different contexts but are equivalent in law can have different asymptotic distributions. We consider Moran's sum of logarithms of uniform spacings statistic which is equal in law to a special case of Bartlett's test statistic for homoscedasticity. Moran's statistic is asymptotically normal and, by accelerating the convergence, we obtain an accurate approximation for small samples. Also, contrary to expectation, we find that the asymptotic chi-square distribution associated with the homoscedasticity test provides as good a basis for approximating the distribution of Moran's statistic. A closer look at likely alternatives leads to a scaled Beta as a more uniformly accurate approximation than either the normal or the chi-square.

Keywords:

• Bartlett statistic
• goodness-of-fit
• uniformity
• approximations