We study Kolmogorov–Smirnov goodness-of-fit tests for evaluating distributional hypotheses where unknown parameters need to be fitted. Following the work of Pollard (1980), 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 2007–2012. When the data relate to estimating the upper 0.1% or higher income shares, the conventional assumption of a Pareto tail distribution cannot be rejected. But the Pareto tail hypothesis is rejected for estimating the top 1.0% or 0.5% income shares at the 5% significance level. A supplement containing proofs and data descriptions is available online.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Social Sciences (miscellaneous)
- Economics and Econometrics
- Statistics, Probability and Uncertainty