Inference about a Common Mean Problem in the Context of Univariate and Multivariate Settings


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Mathematics and Statistics



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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.