Three Essays on Mixture Model and Gaussian Processes

Abstract

This dissertation includes three essays. In the first essay I study the problem of density estimation using normal mixture models. Instead of selecting the ‘right’ number of components in a normal mixture model, I propose an Averaged Normal Mixture (ANM) model to estimate the underlying densities based on model averaging methods, combining normal mixture models with different number of components. I use two methods to estimate the mixing weights of the proposed Averaged Normal Mixture model, one is based on likelihood cross validation and the other is based on Bayesian information criterion (BIC) weights. I also establish the theoretical properties of the proposed estimator and the simulation results demonstrate its good performance in estimating different types of underlying densities. The proposed method is also employed to a real world data set, empirical evidence demonstrates the efficiency of this estimator. The second essay studies short term electricity demand forecasting using Gaussian Processes and different forecast strategies. I propose a hybrid forecasting strategy that combines the strength of different forecasting schemes to predict 24 hourly electricity demand for the next day. This method is shown to provide superior point and overall probabilistic forecasts. I demonstrate the economic utility of the proposed method by illustrating how the Gaussian Process probabilistic forecasts can be used to reduce the expected cost of electricity supply relative to conventional regression methods, and in a decision-theoretic framework to derive an optimal bidding strategy under a stylized asymmetric loss function for electricity suppliers. The third essay studies a non-stationary modeling approach based on the method of Gaussian process regression for crop yields modeling and crop insurance rate estimation. Our approach departs from the conventional two-step estimation procedure and allows the yield distributions to vary over time. I further develop a performance weighted model averaging method to construct densities as mixture of Gaussian processes. This method not only facilitates information pooling but greatly improves the flexibility of the resultant predictive density of crop yields. The simulation results on corp insurance premium rates show that the proposed method compares favorably to conventional two stage estimators, especially when the underlying distributions are non-stationary. I illustrate the efficacy of the proposed method with an application to crop insurance policy selection by insurance companies. I adopt a decision theoretic framework in this exploration and demonstrate that insurance companies can use the proposed method to effectively identify profitable policies under symmetric or asymmetric loss functions.