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Abstract: This paper is concerned with hierarchical data having three levels. The level 1 units are nested in the level 2 units or subclusters which are themselves nested in the level 3 clusters. The model for this data is assumed to fulfill some symmetry assumptions. The level 1 units within each subcluster are exchangeable and a permutation of the subclusters belonging to the same cluster leaves the model unchanged. We are interested in measuring the dependence associated to clusters and subclusters respectively. Two exchangeable Kendall’s tau are proposed as non parametric measures of these two associations and estimators for these measures are proposed. Their asymptotic properties are then investigated under the proposed hierarchical model for the data. These statistics are then used to estimate the intra-class correlation coefficients for data drawn from elliptical hierarchical distributions. Hypothesis tests for the cluster and subcluster effects based on the proposed estimators are developed and their performances are assessed using Pitman efficiencies and a Monte Carlo study. [Copyright &y& Elsevier]

Abstract: We show that under different moment bounds on the underlying variables, bootstrap approximation to the large deviation probabilities of standardized sample sum, based on independent random variables, is valid for a wider zone of , the sample size, compared to the classical normal tail probability approximation. As an application, different notions of efficiency for statistical tests are considered from Bayesian point of view. In particular, efficiency due to Pitman (1938) , Chernoff (1952) , and Bayes risk efficiency due to Rubin and Sethuraman (1965) turn out to be special cases with the choice of the weight function; i.e., prior density times loss. [Copyright &y& Elsevier]

In this paper multivariate extensions of the Friedman and Page tests for the comparison of several treatments are introduced. Related unadjusted and adjusted treatment effect estimates for the multivariate response variable are also found and their properties discussed. The test statistics and estimates are analogous to the traditional univariate methods. In test constructions, the univariate ranks are replaced by multivariate spatial ranks (J. Nonparam. Statist. 5 (1995) 201). Asymptotic theory is developed to provide approximations for the limiting distributions of the test statistics and estimates. Limiting efficiencies of the tests and treatment effect estimates are found in the multivariate normal and t distribution cases. The tests are rotation invariant only, but affine invariant versions can be easily constructed. The theory is illustrated by an example. [Copyright &y& Elsevier]