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

Please use this identifier to cite or link to this item: http://hdl.handle.net/1928/22051

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Title

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

Author(s)

Lin, Yong

Advisor(s)

Christensen, Ronald

Committee Member(s)

Bedrick, Edward

Erhardt, Erik

Sonkson, Michael

Erhardt, Erik

Sonkson, Michael

Department

University of New Mexico. Dept. of Mathematics and Statistics

LC Subject(s)

Linear models (Statistics)

Analysis of covariance.

Asymptotic distribution (Probability theory)

Analysis of covariance.

Asymptotic distribution (Probability theory)

Degree Level

Doctoral

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 Wei’s 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.

Date

December 2012

Subject(s)

Lack-of-fit test

Partial sums of residuals

Asymptotic distribution

Singular covariance matrix

Partial sums of residuals

Asymptotic distribution

Singular covariance matrix

##### Collections

- Statistics [16]
- Statistics Dissertations [9]