Inference about a Common Mean Problem in the Context of Univariate and Multivariate Settings
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Author/Creator ORCID
Date
2023-01-01
Type of Work
Department
Mathematics and Statistics
Program
Statistics
Citation of Original Publication
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Access limited to the UMBC community. Item may possibly be obtained via Interlibrary Loan thorugh a local library, pending author/copyright holder's permission.
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Abstract
This Ph.D. thesis addresses the challenge of validating research hypotheses usingmultiple datasets and focuses on developing ecient testing methods with maximum
discriminatory power to distinguish between true and false hypotheses. This task is
particularly complex when hypotheses are closely positioned, presenting a demanding
scenario. The thesis extensively explores strategies for amalgamating results from
k independent studies that share a common goal, considering both univariate and
multivariate common mean problems with limited information about dispersion parameters.
Local powers of dierent synthesis methods are compared to ascertain their
eectiveness.
For the univariate common mean problem, explicit expressions for local power
are derived for several exact tests, facilitating a comprehensive comparison. The investigation
reveals that a uniform comparison of these tests, irrespective of unknown
variances, is possible for equal sample sizes. Remarkably, the Inverse Normal p-valuebased
exact test emerges as the most eective, and the Modied-F exact test exhibits
a notable advantage among modied distribution-based tests.
The study extends to the multivariate common mean problem, encompassing inference
about a common mean vector from independent multivariate normal populations
with unknown, possibly unequal dispersion matrices. An unbiased estimate of
the common mean vector, along with its asymptotic estimated variance, is proposed
for hypothesis testing and condence ellipsoid construction, applicable to large samples.
Multiple exact test procedures and condence set construction techniques are
meticulously explored, accompanied by a comparative analysis based on local power.
The results reveal that, in the special case of equal sample sizes, local powers are
directly comparable regardless of the unknown dispersion matrices, with Inverse Normal
and Jordan-Kris methods exhibiting superior performance.
The thesis also addresses scenarios where studies contain either univariate or bivariate
features, requiring distinct statistical meta-analysis approaches. Methods for
hypothesis testing in the common mean problem are demonstrated, considering various
strategies for estimating between-studies variability parameters in cases where
homogeneity assumptions are violated. The presented methods are illustrated using
simulated and real datasets.
In summary, this Ph.D. thesis contributes a comprehensive exploration of synthesis
methods for validating research hypotheses in the context of univariate and
multivariate common mean problems. The work showcases the eectiveness of speci
c exact tests and synthesis approaches, providing valuable insights for conducting
statistical meta-analysis across diverse scenarios.