Likelihood-based methods for constrained parameter problems

Abstract

Truncated observations for some applications and parameters with a certain kind of constraints may provide a kind of prior information for the data analysis. Such information, if integrated into the scientific study, can significantly improve the result of statistical inferences. In order to sufficiently utilize such information, these truncations or constraints are taken into consideration in the modeling process. This thesis, therefore, aims to analyze truncated normal data and to study the constrained parameter problems.

In practice, the normal distribution is widely used to model continuous data. However, when the data fall in a certain interval, truncated normal distribution becomes a better choice. Through stochastically representing the normal random variable as a mixture of a truncated normal random variable and its complementary random variable, Chapter 2 proposes two new expectation-maximization (EM) algorithms to calculate maximum likelihood estimates of parameters in truncated normal distribution. Furthermore, in the analyses of two real datasets based on Akaike information criterion (AIC) and Bayesian information criterion (BIC), it is found that the truncated normal distribution performs better than the half normal, the folded normal and the folded normal slash distributions.

Although Type I multivariate zero-inflated Poisson (ZIP) distribution has been recently proposed to model zero inflated correlated multivariate discrete data by Liu and Tian (2015), the statistical methods for this multivariate distribution with constrained parameters are still lacking. Chapter 3 proposes an SR-based EM algorithm (Dempster et al., 1977) and a Q-based EM algorithm aided by the De Pierro algorithm (De Pierro, 1995) for the constrained multivariate ZIP models after studying two constrained types.

Generalized linear model (GLM) with canonical link function is a flexible and useful generalization of the ordinary linear regression. However, no thorough studies on the constrained estimation problem in GLM have been done so far. According to the Karush-Kuhn-Tucker conditions, Chapter 4 derives two asymptotic properties of the constrained estimators. Meanwhile, via transferring the constrained optimization problem of maximizing a log-likelihood function to the problem of maximizing a separable surrogate function with a diagonal Hessian matrix subject to box constraints, the constrained optimization problem is now equivalent to separately maximizing several one dimensional concave functions with a lower bound and an upper bound and has therefore an explicit solution. Furthermore, after this transformation, a modified De Pierro (DP) algorithm is developed to calculate the maximum likelihood estimates (MLE) of the regression coefficients subject to linear or box inequality restrictions. Lastly, the analyses of real datasets and simulations are conducted to evaluate the proposed methods in this thesis.