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1 
Ftests in partially balanced and unbalanced mixed linear modelsUtlaut, Theresa L. 11 February 1999 (has links)
This dissertation considers two approaches for testing hypotheses in
unbalanced mixed linear models. The first approach is to construct a design with
some type of structure or "partial" balance, so that some of the optimal properties of
a completely balanced design hold. It is shown that for a particular type of partially
balanced design certain hypothesis tests are optimal. The second approach is to
study how the unbalancedness of a design affects a hypothesis test in terms of level
and power. Measures of imbalance are introduced and simulation results are
presented that demonstrate the relationship of the level and power of a test and the
measures.
The first part of this thesis focuses on error orthogonal designs which are a
type of partially balanced design. It is shown that with an error orthogonal design
and under certain additional conditions, ANOVA Ftests about certain linear
combinations of the variance components and certain linear combinations of the
fixed effects are uniformly most powerful (UMP) similar and UMP unbiased. The
ANOVA Ftests for the variance components are also invariant, so that the tests are
also UMP invariant similar and UMP invariant unbiased. For certain simultaneous
hypotheses about linear combinations of the fixed effects, the ANOVA Ftests are
UMP invariant unbiased.
The second part of this thesis considers a mixed model with a random
nested effect, and studies the effects of an unbalanced design on the level and
power of a hypothesis test of the nested variance component being equal to zero.
Measures of imbalance are introduced for each of the four conditions necessary to
obtain an exact test. Simulations are done for two different models to determine if
there is a relationship between any of the measures and the level and power for both
a naive test and a test using Satterthwaite's approximation. It is found that a
measure based on the coefficients of the expected mean squares is indicative of
how a test is performing. This measure is also simple to compute, so that it can
easily be employed to determine the validity of the expected level and power. / Graduation date: 1999

2 
Introduction to Probability TheoryChen, YongYuan 25 May 2010 (has links)
In this paper, we first present the basic principles of set theory and combinatorial analysis which are the most useful tools in computing probabilities. Then, we show some important properties derived from axioms of probability. Conditional probabilities come into play not only when some partial information is available, but also as a tool to compute probabilities more easily, even when partial information is unavailable. Then, the concept of random variable and its some related properties are introduced. For univariate random variables, we introduce the basic properties of some common discrete and continuous distributions. The important properties of jointly distributed random variables are also considered. Some inequalities, the law of large numbers and the central limit theorem are discussed. Finally, we introduce additional topics the Poisson process.

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