Estimating functions have been widely used for parameter
estimation in various statistical problems. Regular estimating
functions produce parameter estimators which have desirable
properties, such as consistency and asymptotic normality. In
quasi-likelihood inference, an important example of estimating
functions, correct specification of the first two moments of the
underlying distribution leads to the information unbiasedness, which
states that two forms of the information matrix: the negative
sensitivity matrix (negative expectation of the first order
derivative of an estimating function) and the variability matrix
(variance of an estimating function) are equal, or in other words,
the analogue of the Fisher information is equivalent to the Godambe
information. Consequently, the information unbiasedness indicates
that the model-based covariance matrix estimator and sandwich
covariance matrix estimator are equivalent. By comparing the
model-based and sandwich variance estimators, we propose information
ratio (IR) statistics for testing model misspecification of
variance/covariance structure under correctly specified mean
structure, in the context of linear regression models, generalized
linear regression models and generalized estimating equations.
Asymptotic properties of the IR statistics are discussed. In
addition, through intensive simulation studies, we show that the IR
statistics are powerful in various applications: test for
heteroscedasticity in linear regression models, test for
overdispersion in count data, and test for misspecified variance
function and/or misspecified working correlation structure.
Moreover, the IR statistics appear more powerful than the classical
information matrix test proposed by White (1982).
In the literature, model selection criteria have been intensively
discussed, but almost all of them target choosing the optimal mean
structure. In this thesis, two model selection procedures are
proposed for selecting the optimal variance/covariance structure
among a collection of candidate structures. One is based on a
sequence of the IR tests for all the competing variance/covariance
structures. The other is based on an ``information discrepancy
criterion" (IDC), which provides a measurement of discrepancy
between the negative sensitivity matrix and the variability matrix.
In fact, this IDC characterizes the relative efficiency loss when
using a certain candidate variance/covariance structure, compared
with the true but unknown structure. Through simulation studies and
analyses of two data sets, it is shown that the two proposed model
selection methods both have a high rate of detecting the
true/optimal variance/covariance structure. In particular, since the
IDC magnifies the difference among the competing structures, it is
highly sensitive to detect the most appropriate variance/covariance
structure.
| Identifer | oai:union.ndltd.org:WATERLOO/oai:uwspace.uwaterloo.ca:10012/4614 |
| Date | January 2009 |
| Creators | Zhou, Qian |
| Source Sets | University of Waterloo Electronic Theses Repository |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
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