In this master thesis, we explore the probability theory, statistical inference and numerical computation of discrete compound Poisson (DCP) distribution. In particular, we do a very comprehensive literature review of DCP distributions and its applications in related statistical models of count data fields, and especially, we discuss penalized generalized linear model of count data regression.The discrete compound Poisson distributions have the probability generating function in the form of the following: The famous Feller’s characterization of the compound Poisson states that a discrete distribution is compound Poisson if and only if its distribution is discrete infinitely divisible. This is a special case of Levy-Khinchine formula. When the{ai}i=1∞, may take negative values and the sum is absolutely convergent, it is called pseudo discrete compound Poisson distribution.In the first chapter, we introduce an important tool (probability generating function and Fourier transform) as preliminaries and improve the flawed proof of Feller’s characterization, and then we give a short introduction of variable selection method about Lasso and generalization. We close this chapter with the infinitely divisibile prior distribution in Bayesian Lasso and we envisages appropriate zero-inflated distribution as prior distribution which obtains the nonzero sparse estimation of coefficients. The chapter Ⅱ discusses characterizations of DCP distribution(process) with ten methods to prove the probability mass function are given in Appendix, and we give over a hundred kinds of special cases or sub-families of DCP distribution which are listed in a table with references. We use Stein-Chen method and operator semigroup method to obtain the upper bound of the total variation between a sum of independent discrete r.v. and a related discrete compound Poisson r.v., and use row sum in random triangular array to approximate discrete compound Poisson distribution. Chapter Ⅳ studys statistics, parameters estimation, FFT of DCP probability mass. Chapter Ⅴ firstly uses cumulants estimation and Fourier transform estimation to actuarial claim data with zero-inflated and overdispersion properties, then compares its Kolmogorov-Smimov test and Chi-squared test. We give a theorem that a set of count data obeys discrete pseudo compound Poisson distribution if its. probability of zero is larger than the probability of nonzero. Further more, we use this zero-inflated property of pseudo discrete compound Poisson with adding virtual frequency techniques; we get an algorithm to fit any discrete distributions. Chapter V also discusses count GLM related to the DCP distribution and use penalized estimation to select important regression variables. In particular, we consider the Elastic net estimates of negative binomial regression, and we give a necessary and sufficient condition(like Karush-Kuhn-Tucker conditions) for non-zero(zero) coefficient estimates. Using a spider count data, we analysis this real example by negative binomial regression with MLE, Lasso, Elastic net penalties. Next, we set forth the survival functions in discrete frailty model and cured rate models (or long term survivor models with competing causes) which are derived from some DCP distributions. In the last section, we look forword to the future study that mixed Poisson distribution to approximate any discrete distribution, and states the problem of variable selection in mixture components. Due to the complexity of the mixture, it results the high-dimentional problem.
This thesis develops models and associated Bayesian inference methods for flexible univariate and multivariate conditional density estimation. The models are flexible in the sense that they can capture widely differing shapes of the data. The estimation methods are specifically designed to achieve flexibility while still avoiding overfitting. The models are flexible both for a given covariate value, but also across covariate space. A key contribution of this thesis is that it provides general approaches of density estimation with highly efficient Markov chain Monte Carlo methods. The methods are illustrated on several challenging non-linear and non-normal datasets. In the first paper, a general model is proposed for flexibly estimating the density of a continuous response variable conditional on a possibly high-dimensional set of covariates. The model is a finite mixture of asymmetric student-t densities with covariate-dependent mixture weights. The four parameters of the components, the mean, degrees of freedom, scale and skewness, are all modeled as functions of the covariates. The second paper explores how well a smooth mixture of symmetric components can capture skewed data. Simulations and applications on real data show that including covariate-dependent skewness in the components can lead to substantially improved performance on skewed data, often using a much smaller number of components. We also introduce smooth mixtures of gamma and log-normal components to model positively-valued response variables. In the third paper we propose a multivariate Gaussian surface regression model that combines both additive splines and interactive splines, and a highly efficient MCMC algorithm that updates all the multi-dimensional knot locations jointly. We use shrinkage priors to avoid overfitting with different estimated shrinkage factors for the additive and surface part of the model, and also different shrinkage parameters for the different response variables. In the last paper we present a general Bayesian approach for directly modeling dependencies between variables as function of explanatory variables in a flexible copula context. In particular, the Joe-Clayton copula is extended to have covariate-dependent tail dependence and correlations. Posterior inference is carried out using a novel and efficient simulation method. The appendix of the thesis documents the computational implementation details.