Rao's 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.