# Contributions to linear models : lack-of-fit test and linear model with singular covariance matrices

## Publication Date

1-31-2013

## Abstract

Linear models are statistical models that are linear in their parameters. This class of models include traditional regression, ANOVA, ACOVA, mixed models and even many time series models. They can be extended into generalized linear models in which case the parameters are still linear, but they are not linearly associated with the dependent variables. This dissertation contributes in two directions. First, it proposes and studies new lack-of-fit tests. Su and Wei (1991) proposed a lack-of-fit test based on partial sums of residuals. They computed P values using an unusual bootstrapping simulation. However, the simulation can not be performed for even moderate numbers of predictor variables because it is prohibitively time consuming. I examine the nature of their bootstrap simulation and argue that it reduces the power of Su and Weis test. I modify their test for linear models and propose two lack-of-fit tests based on partial sums of residuals. I find the non-normal limiting distributions for both tests and small sample corrections that enable more precise calculation of 0.05 cut-offs. Empirical sizes and powers are studied for both tests in small samples. In the second contribution, I studied the linear model with singular covariance matrix. In these models, frequently there exists estimable functions of that are known with probability 1. Traditional methods of analysis employ a psuedo-covariance matrix that gives BLUEs and tests that are appropriate for the actual covariance matrix V . Contrary to traditional methods of adjusting V , I decompose into known and unknown parts and adjust X to allow estimation and testing of the unknown part of . Specifically, I adjust this model, Y = X + e, to get an equivalent model, Y − X 0 = Xv + e, where X 0 is a known vector, then perform estimation and tests on this equivalent model. The equivalence of the models is studied.

## Degree Name

Statistics

## Level of Degree

Doctoral

## Department Name

Mathematics & Statistics

## First Committee Member (Chair)

Ronald Christensen

## Second Committee Member

Edward John Bedrick

## Third Committee Member

Erik Barry Erhardt

## Fourth Committee Member

Michael D. Sonksen

## Language

English

## Keywords

Linear models (Statistics), Analysis of covariance, Asymptotic distribution (Probability theory)

## Document Type

Dissertation

## Recommended Citation

Lin, Yong. "Contributions to linear models : lack-of-fit test and linear model with singular covariance matrices." (2013). https://digitalrepository.unm.edu/math_etds/61