Abstract:

In multivariate regression, researchers are interested in modeling a correlated multivariate response variable as a function of covariates. The response of interest can be multidimensional; the correlation between the elements of the multivariate response can be very complex. In many applications, the association between the elements of the multivariate response is typically treated as a nuisance parameter. The focus is on estimating efficiently the regression coefficients, in order to study the average change in the mean response as a function of predictors. However, in many cases, the estimation of the covariance and, where applicable, the temporal dynamics of the multidimensional response is the main interest, such as the case in finance, for example. Moreover, the correct specification of the covariance matrix is important for the efficient estimation of the regression coefficients. These complex models usually involve some parameters that are static and some dynamic. Until recently, the simultaneous estimation of dynamic and static parameters in the same model has been difficult. The introduction of particle MCMC algorithms has allowed for the possibility of considering such models. In this thesis, we propose a general framework for jointly estimating the covariance matrix of multivariate data as well as the regression coefficients. This is done under different settings, for different dimensions and measurement scales.

Abstract:

In this talk, we propose a robust model fitting algorithm for Generalized Partial Linear Partial Additive Models (GAPLMs), which is a hybrid of the widely-used Generalized Linear Models (GLMs) and Generalized Additive Models (GAMs). The traditional model fitting algorithms are mainly based on likelihood. However, those fits can be severely distorted by the presence of a small portion of atypical observations (also known as "outliers"), which deviate from the assumed model. As a result, the fits become close to those outliers making them not seem atypical. In order to solve this problem, we developed a model fitting algorithm which is resistant to the effect of outliers. To fit the "partial linear partial additive" styled model, our method involves backfitting algorithm and generalized Speckman estimator. To achieve a robust fit, we applied the robust weights derived from robust quasi-likelihood equations proposed by Cantoni and Ronchetti 2001, instead of the likelihood based weights, in generalized local scoring algorithm. To compare the our model fitting performance with the non-robust fit given by the R function gam::gam(), we operated a simulation study and applied the two fitting methods on an example of real dataset. It has been shown in our studies that our robust model fitting algorithm can effectively resist the effect of atypical observations and identify outliers by comparing the robust fitted values with the observed response variable.