We study Kolmogorov-Smirnov goodness of ﬁt tests for evaluating distributional hypotheses where unknown parameters need to be ﬁtted. Following work of Pollard (1979), our approach uses a Cramér-von Mises minimum distance estimator for parameter estimation. The asymptotic null distribution of the resulting test statistic is represented by invariance principle arguments as a functional of a Brownian bridge in a simple regression format for which asymptotic critical values are readily delivered by simulations. Asymptotic power is examined under ﬁxed and local alternatives and ﬁnite sample performance of the test is evaluated in simulations. The test is applied to measure top income shares using Korean income tax return data over 2007 to 2012. When the data relate to the upper 0.1% or higher tail of the income distribution, the conventional assumption of a Pareto tail distribution cannot be rejected. But the Pareto tail hypothesis is rejected for the top 1.0% or 0.5% incomes at the 5% signiﬁcance level.
Cho, Jin Seo; Park, Myung-Ho; and Phillips, Peter C.B., "Minimum Distance Testing and Top Income Shares in Korea" (2015). Cowles Foundation Discussion Papers. 2442.