Characterizing and approximating scale mixtures

The purpose of this study is to identify and approximate infinite scale mixtures of normals, infinite scale mixture of uniforms and infinite scale mixtures of exponentials. We present a new method for approximating an infinite scale mixture with a known mixing measure by a scale mixture with a finite number of terms. The method is based on discretizing the mixing measure. We wish to minimize the number of terms, which depends on the mixing measure, used in this approximation process. The first part of the thesis will focus on scale mixture of normals in the univariate and multivariate setting. The new method will be used to approximate invariant and multivariate variance mixtures of normals with mean 0. Two different ways of applying the method in the multivariate case are introduced. Several results related to variance mixtures of normals are proved. Two different ways of estimating the mixing measure of the variance mixture of normals based on a sample are introduced. The first method is called the relative rate of change method, RRC, and the second method is called UNMIX. The estimated mixing measure from the UNMIX method is used to fit the data. Variance-mean mixtures are presented through Barndorff-Nielsen dependence structure. The approximation method is used to approximate variance-mean mixtures in the independent case. Infinite scale mixtures of uniforms are characterized and the new method is used to approximate them. Similarly, infinite scale mixture of exponentials are characterized and approximated by the new method. The performance of the method depends on what are we mixing and the percentage of the mixing measure in the tail and near the origin.