The purpose of this paper is to construct a linear prediction model for a three-stage hierarchical data structure and develop a Bayesian procedure to estimate parameters. More specifically, the regression coefficient vectors in this model are assumed to be exchangeable within a block in the de Finetti's sense, and the block parameters are also assumed to be exchangeable. We employed an E-M algorithm in order to obtain estimates for the hyperparameters of the model. Then, conditional on these estimates, we estimated the rest of the parameters which are relevant for prediction.
Finally, we applied this estimation procedure to the entrance examination data and compared its predictive efficiency with other possible prediction models. We found that the proposed model produced the best root mean squared errors, although the difference from other models was minimal.

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In this paper we derive the risk function of the unrestricted and the restricted Zellner's estimators in the SUR equations model used in Revankar (1974, 1976) under the quadratic loss function when the model is misspecified. Using numerical calculations, we compare the risk performance of the OLS and Zellner's estimators. As the result of the analysis, the misspecification in the model influences the risk performance of estimators greatly.

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The testing problems of multivariate several linear relationships with replication are considerd. The asymptotic distributions of Basu's test statistics under the null hypotheses are obtained for both functional and structural relationships, and these are shown to the same for both relationships.

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Simultaneous confidence intervals for multiple comparisons in a generalized multivariate analysis of variance model with covariance structures are discussed. We consider the model with positive definite, serial, and uniform covariance structures. The simultaneous confidence intervals using the modified second approximation procedure are given. In related discussion, asymptotic expansions for the joint probability of bivariate Hotelling T²-type variates are derived under the model with the covariance structures.

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This paper discusses estimation of the kurtosis parameter, a key parameter in elliptical distributions. Asymptotic properties of the usual consistent estimator of kurtosis parameter in elliptical distributions are given by the perturbation method. Estimators with bias correction based on an asymptotic expansion of the expectation of the usual estimator are also given. Finally, the bias and the mean square error for the estimators are investigated by Monte Carlo simulation for some selected parameters.

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Higher order asymptotic aspects of the jackknife-t method are studied. Emphasis is placed upon constructing higher order accurate confidence intervals for parameters which are smooth functions of multivariate means. Explicit formulae for two-term Edgeworth corrections for both cumulative distribution functions and percentiles are provided. These formulae can be automatically evaluated given specific distributions. Jackknife-t confidence intervals having coverage error of O(n^-3/2), n being the sample size, are obtained by twice inverting certain Edgeworth expansion. A numerical example is given in the case of estimating the coefficient of variation from normal populations. Bootstrap intervals are also discussed.

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We study the effect on residual autocorrelations in the presence of atypical observations for the AR (1) process. Based on the empirical influence function of the portmanteau statistic, a measure of influence is derived. A confirmatory procedure coupling robust estimation with Monte-Carlo test is then formulated to assess the magnitude of the sample statistics. An example illustrates application of the methodology.

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This paper deals with the likelihood ratio test for a hypothesis concerning the relevance of a subset of variables from each of two given variable sets in a canonical correlation analysis (CCA). This hypothesis can be considered as having two conditions for the covariance structure. Assuming that one of these two conditions holds, we obtain the exact moments of the likelihood ratio criterion and the asymptotic expansions of the power function of the likelihood ratio test.

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Takeuchi [37], Takeuchi and Akahira [38] and Pfanzagl [27] among others proved that any first-order efficient estimators are second-order efficient. Many other authors e. g., Ghosh [15], have conjectured that any third-order efficient estimators are also fourth-order efficient. Based on the concentration probability of estimators about a true parameter, this paper gives a positive answer to the conjecture in a curved exponential family with multi-structural parameters. It is seen that choice of bias-correction factors is critical.

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Discrete mixtures of exponential random variables X₁ and X₂ are identified in terms of (i) relations between the best predictor of max (X₁, X₂) given min (X₁, X₂) and the failure rate of the distribution, and (ii) relations between conditional expectation of some appropriate functions of order statistics on some others and the failure rate of the distribution.

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