A New Statistic for Testing an Assumed Distribution

Summary

A new statistic, T, for testing an assumed distribution of the form $\frac{1}{\sigma }f\left( {\frac{{x - \mu }}{\sigma }} \right)$ is proposed. This statistic is the ratio of V. the maximum likelihood estimator or modified maximum likelihood estimator (Tiku, 1967, 1973) of O calculated from a censored sample, to the maximum likelihood estimator calculated from the whole sample. T is both location and scale invariant. The asymptotic distribution of T is normal  and for some populations the distribution of T is exactly beta. T is, in general, easy to compute and has good power properties. The statistic T can also be generalized to multi-sample situations in a straightforward fashion. However, in the use of T one has to have a priori knowledge whether the alternative distribution is skewed or symmetric.