Rao\u27s Quadratic Entropy and Some New Applications

Many problems in statistical inference are formulated as testing the diversity of populations. The entropy functions measure the similarity of a distribution function to the uniform distribution and hence can be used as a measure of diversity. Rao (1982a) proposed the concept of quadratic entropy. Its concavity property makes the decomposition similar to ANOVA for categorical data feasible. In this thesis, after reviewing the properties and providing a modification to quadratic entropy, various applications of quadratic entropy are explored. First, analysis of quadratic entropy with the suggested modification to analyze the contingency table data is explored. Then its application to ecological biodiversity is established by constructing practically equivalent confidence intervals. The methods are applied on a real dinosaur diversity data set and simulation experiments are performed to study the validity of the intervals. Quadratic entropy is also used for clustering multinomial data. Another application of quadratic entropy that is provided here is to test the association of two categorical variables with multiple responses. Finally, the gene expression data inspires another application of quadratic entropy in analyzing large scale data, where a hill-climbing type iterative algorithm is developed based on a new minimum quadratic entropy criterion. The algorithm is illustrated on both simulated and real data

Minimum Distance Testing and Top Income Shares in Korea

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