Correspondence analysis is often used for analysis of the relevance of the category data of a two-way contingency table. The notion of dimensionality in correspondence analysis has been introduced in an analogy with canonical correlation analysis. It reveals with the dimension in which space the relationship can be contracted. A criterion for estimating the dimensionality has been proposed by Ogura and Fujikoshi (2013), by modifying a formal AIC. In this paper, first we note that the criterion is related to an estimator of the expected prediction error in least squares. Then, we study asymptotic behavior of the criterion by simulation experiments. Through simulation experiments and an application to a real data, it is shown that our criterion works well.

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In this paper, first we consider the chi-square approximation of Takemura & Sheena (2005) and extend it to derive an approximate distribution of higher numerical precision. Furthermore, we demonstrate that this extension provides almost the same result as that found by Sugiyama (1972) for the distribution of the largest eigenvalue. We then consider the confidence interval of each eigenvalue and perform a numerical comparison between Takemura and Sheena's chi-square approximation and the approximated distribution found with the extended method to demonstrate that the latter has higher accuracy. Next, we consider the simultaneous confidence interval for all population eigenvalues discussed by Anderson (1965). Anderson employed the chi-square approximation of the largest and of the smallest eigenvalues of a Wishart matrix to construct the simultaneous confidence interval. This method can be interpreted as a different viewpoint of employing Takemura and Sheena's results, namely that the distributions of the largest and of the smallest eigenvalue of a Wishart matrix can be approximated with a chi-square distribution. Accordingly, an extension of Takemura & Sheena (2005) can express the simultaneous confidence interval of all population eigenvalues obtained by the procedure of Anderson (1965). When the proposed approximation distribution is used, one must establish the confidence intervals using all the estimate eigenvalues of the population; even though the issue of correcting the bias in the estimates must subsequently be addressed, ultimately, our proposed simultaneous confidence interval is reasonably good in terms of having type I errors that are closer to 5%.

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In this paper, we consider some procedures of testing equality of mean vectors and of profile analysis under elliptical populations having correlation among them. Hotelling's T^2 type statistic is important for testing equality of mean vectors and is discussed in normal and non-normal populations by many authors. Test for equality of mean vectors in multivariate normal populations which have correlation among populations is introduced by Morrison (2005). This test statistic is usually called Paired T^2 statistic, which is Hotelling's T^2 type statistic. In order to consider testing the equality of the mean vectors and construct the simultaneous confidence intervals, we derive the approximate upper percentiles for the distribution of the paired T^2 statistic. Moreover, we consider the profile analysis using Hotelling's T^2 type statistics. Profile analysis in elliptical populations is discussed by Okamoto et al. (2006) and so on. We derive the approximate upper percentiles in elliptical populations with correlation. Finally, the accuracy of the approximate values for Paired T^2 type statistics and Hotelling's T^2 type statistic for profile analysis is investigated by Monte Carlo simulations for selected values of parameters.

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