| The concept of dispersion associated with a probability vector is an important issue both statistically and within other scientific disciplines. Several functions to quantify dispersion have been proposed, including the familiar Shannon index (entropy) and the Gini index (concentration). Applications where measures of dispersion are important include economic studies on wealth and income, ecology studies in species diversity, and sociological studies on occupational status. Dispersion measures defined on probability vectors for nominal categories are considered. We limit our consideration to a large family of dispersion measures satisfying a set of general constraints. Of interest are the hypothesis of equal dispersion for all measures in this family and the hypothesis that dispersions associated with one probability vector are greater than or equal to the dispersions associated with the other. We obtain maximum likelihood estimates of the probability vector parameters under both hypotheses and show that these estimators are consistent under their respective constraints. The likelihood-ratio test statistic for testing equality of dispersion against a one sided alternative is computed and its asymptotic null distribution is derived. The asymptotic null distribution is not in closed form; thus, approximations are necessary to perform our test. Two different testing approaches are considered: using an approximate fixed, known, critical value to perform the test and using the bootstrap method to obtain a p-value for our test statistic. Through simulation studies, these approaches are shown to work well. In addition, we show that for the special case of distinct component values within each of the two probability vectors, the asymptotic null distribution of our test statistic has a chi bar squared distribution. Simulations are used to investigate the power of our test and to compare our test to other tests of dispersion. Several data sets are analyzed using the methods developed in this thesis. |
