Professor Kai-Sheng Song invites you to attend the

Doctoral Dissertation Defense of Ifeanyichukwu Valentine Ukenazor

"Scale Invariant Equations and Its Modified EM Algorithm for Estimating A Two-Component Mixture Model"

ABSTRACT:

A two-component mixture model is a statistical model for representing a probability distribution that is a combination of two distinct sub-populations. It is widely used in many applications to model the data-generating process that often exhibits bimodal distributions. The component distributions of such finite mixture models can be specified either parametrically or nonparametrically or semi-parametrically. Among parametric finite mixture models, Gaussian mixtures are the most popular choice in applications because they are relatively easy to interpret and have well-studied theoretical properties. Despite the usual smoothness of typical parametric finite mixture models, the mixing of components makes their likelihood functions analytically intractable, consequently model parameters are often estimated by the Expectation-Maximization (EM) algorithm. However, it is well-known that the EM algorithm is very sensitive to starting values and may converge to local extrema, especially for small sample sizes, thus requiring good initial estimators.

In this talk, we propose a novel two-component mixture model: the first component is the three-parameter generalized Gaussian distribution (GGD) and the second component is a new three-parameter family of positive densities on the real line. The novelty of our model is that we allow the two components to have totally different parametric families of distributions with asymmetric tails of the mixture density. We extend the scale invariant variable fractional moments (SIVFM) method proposed by Song (2006) for the GGD to the parameter estimation of our mixture model. We show that the SIVFM population and sample equations for the second component share very similar desirable global properties such as convexity and unique global roots as those for the GGD given in Song (2006). The two-component mixing of these properties makes the SIVFM mixture population and estimation equations well-behaved resulting in easy to compute SIVFM estimators without the issue with starting values. The asymptotic results such as consistency and limiting distribution of the proposed estimators are presented. Furthermore, SIVFM estimators can also serve as a consistent initial estimator for the EM algorithm leading to improved accuracy of the EM algorithm. These algorithms are applied to the analysis of the average amount of precipitation (rainfall) for each of 70 United States (and Puerto Rican) cities clearly demonstrating the bimodal distribution of the estimated mixture density.