Linear algebra.- Random vectors.- Gamma, Dirichlet, and F distributions.- Invariance.- Multivariate normal.- Multivariate sampling.- Wishart distributions.- Tests on mean and variance.- Multivariate regression.- Principal components.- Canonical correlations.- Asymptotic expansions.- Robustness.- Bootstrap confidence regions and tests.

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Theory of multivariate statistics

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1600-6143.2011.03656.x

Pregnancy Outcomes in Kidney Transplant Recipients: A Systematic Review and Meta-Analysis

Generalized pivotal quantities (GPQs) and generalized confidence intervals (GCIs) have proven to be useful tools for making inferences in many practical problems. Although GCIs are not guaranteed to have exact frequentist coverage, a number of published and unpublished simulation studies suggest that the coverage probabilities of such intervals are sufficiently close to their nominal value so as to be useful in practice. In this article we single out a subclass of generalized pivotal quantities, which we call fiducial generalized pivotal quantities (FGPQs), and show that under some mild conditions, GCIs constructed using FGPQs have correct frequentist coverage, at least asymptotically. We describe three general approaches for constructing FGPQs—a recipe based on invertible pivotal relationships, and two extensions of it—and demonstrate their usefulness by deriving some previously unknown GPQs and GCIs. It is fair to say that nearly every published GCI can be obtained using one of these recipes. As an inte...

https://www.jstor.org/stable/info/30047454

Fiducial Generalized Confidence Intervals

This paper develops new methodology, together with related theories, for combining information from independent studies through confidence distributions. A formal definition of a confidence distribution and its asymptotic counterpart (i.e., asymptotic confidence distribution) are given and illustrated in the context of combining information. Two general combination methods are developed: the first along the lines of combining p-values, with some notable differences in regard to optimality of Bahadur type efficiency; the second by multiplying and normalizing confidence densities. The latter approach is inspired by the common approach of multiplying likelihood functions for combining parametric information. The paper also develops adaptive combining methods, with supporting asymptotic theory which should be of practical interest. The key point of the adaptive development is that the methods attempt to combine only the correct information, downweighting or excluding studies containing little or wrong information about the true parameter of interest. The combination methodologies are illustrated in simulated and real data examples with a variety of applications.

/pdf/combining-information-from-independent-sources-through-x265lwcy2j.pdf

Combining information from independent sources through confidence distributions

https://aasldpubs.onlinelibrary.wiley.com/doi/pdf/10.1002/lt.23416

Pregnancy outcomes of liver transplant recipients: a systematic review and meta-analysis.

SUMMARY The problem of interval estimation of the common mean of several normal populations when the variances are unknown and unequal is considered. Some new confidence intervals are proposed. One of them is centered at the well-known Graybill-Deal estimator of the common mean. These new intervals and an existing exact confidence interval are numerically compared with respect to their expected lengths. Based on comparison studies, some recommendations are made regarding the choice of the intervals to be used in applications. The problem of combining the results of independent investigations for the purpose of a common objective has received considerable attention in the literature. In particular, it may be necessary to combine the data sets collected by independent agencies to estimate or to test the common parameter of interest. This problem arises, for example, when two or more independent agencies are involved in measuring the effect of a new drug, when several measuring instruments are used to measure the products produced by the same production process to assess the average quality, or when different laboratories are used to measure the amount of toxic waste in a river. Meier (1953) gave an illustrative example in which four experiments are used to estimate the mean percentage of albumin in the plasma protein of normal human subjects. Other examples can be found in the recent papers by Eberhardt, Reeve, and Spiegelman (1989) and Skinner (1991). The former paper gives three examples each of which is concerned with estimating a chemical substance in nonfat milk powder by combining the results of alternative analytical methods; the latter gives examples related to clinical trials and determining the density of nitrogen using different laboratory methods. In these situations, the variances are typically unequal due to the variation in different methods or measuring instruments. If it is assumed that the samples collected by independent studies are from normal populations, then the problem of interest may be to estimate or test the common mean A of these populations.

Exact Confidence Intervals for the Common Mean of Several Normal Populations

Suppose that there are independent samples available from several multivariate normal populations with the same mean vector m but possibly different covariance matrices. The problem of developing a confidence region for the common mean vector based on all the samples is considered. An exact confidence region centered at a generalized version of the well-known Graybill-Deal estimator of m is developed, and a multiple comparison procedure based on this confidence region is outlined. Necessary percentile points for constructing the confidence region are given for the two-sample case. For more than two samples, a convenient method of approximating the percentile points is suggested. Also, a numerical example is presented to illustrate the methods. Further, for the bivariate case, the proposed confidence region and the ones based on individual samples are compared numerically with respect to their expected areas. The numerical results indicate that the new confidence region is preferable to the single-sample versions for practical use.



Supposons qu'il y a des echantillons independants provenant de plusieurs populations a plusieurs variables normales avec le měme vecteur moyen m mais possiblement differentes matrices des covariances. Cette article considere le probleme de developper une region de confiance pour le vecteur moyen commun basee sur tous les echantillons. Une region de confiance exacte centree sur un version generalisee du fameux estimateur de m de Graybill-Deal est developpee et une procedure de comparaisons multiples basee sur cette region de confiance est evoquee. Les points percentiles necessaires a la construction de la region de confiance sont donnes pour le cas ou il y a deux echantillons. Dans le cas ou il y a plus de deux echantillons, une methode pratique d'approximation des points percentiles est suggeree. Aussi, un exemple numerique est presente afin d'illustrer les methodes. De plus, pour le cas bidimensionnel, la region de confiance proposee et celles basees sur des echantillons individuels sont comparees numeriquement quant a leur aires probables. Les resultats numeriques indiquent que la nouvelle region de confiance est preferable aux versions d'echantillon unique pour un usage pratique.

Scott M. Jordan

Papers

Exact Confidence Intervals for the Common Mean of Several Normal Populations

Confidence regions for the common mean vector of several multivariate normal populations

Efficacy of intramuscular chondroitin sulfate and compounded acetyl-d-glucosamine in a positive controlled study of equine carpitis