We adapt the local influence method to linear discriminant analysis for the purpose of investigating the influence of observations. A simultaneous perturbation on all observations coming from two populations is considered. We study the curvatures and the associated direction vectors of the surface formed by the perturbed maximum likelihood estimators of parameters of interest, in addition to the direction vector of the maximum slope. We show that the influence function method gives essentially the same information as the direction vector of the maximum slope. A numerical example illustrates that the local influence method gives valuable information about influential observations and outliers, even when the influence function method and the case deletion method are not adequate.

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The exact distribution of Hartley's Fmax statistic under heterogeneity of variances with or without unequal sample sizes is given. This exact distribution can now provide a statistical test for several variances under unbalanced design of the Analysis of Variance (ANOVA). This distribution under heterogeneity of variances can provide the power of the test with or without unequal sample sizes.
The relaxation of the conditions for the equal sample sizes and for homogeneity of variances will require an astronomical number of statistical tables which cannot practically be shown. We therefore provide a Fortran program¹ which calculates, to a high accuracy, the upper tail probabilities, probability points and probability densities for Fmax statistic under heterogeneity or homogeneity of variances and with or without unequal sample sizes. Usually the accuracy of the calculation is 13 to 14 digits for double precision arithmetic.
The probability density functions under heterogeneity or homogeneity of variances and with or without unequal sample sizes are shown.

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In maximum penalized likelihood estimation, approaches of cross-validation (CV) are often useful in selecting a smoothing parameter. The CV score based on squared-error criterion behaves more badly than the likelihood-based score. However, it is expensive to calculate the likelihood-based score. Hence we propose a method for simple calculation of this score. The simple calculation is derived as an analogue of the deletion lemma in ordinary or penalized least squares, and is shown to be related to the one-step approximation to the estimates of parameters for the Newton-Raphson method. Our method is applied to binary data from some case studies in the context of logistic regression. It is illustrated that the simple calculation method well behaves and gives a good approximation to the likelihood-based score calculated by the delete-one method.

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There have been proposed a lot of models and procedures derived from different approaches to analyses of asymmetric data. In this paper, we are interested in a particular approach that is based on decomposition of asymmetric matrix in the context of linear algebra. We focus on two procedures, and consider some conditions to derive line structure from asymmetric matrix through those procedures. Four theorems are presented and an illustrative example is given.

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In this paper, we discuss an analysis for statistical tables with nondisclosure cells when the data are presented in tabular form. Inaba and Iwasaki (1996, 1997) proposed procedures with the aim of imputing the values for nondisclosure cells. On the other hand, the major objective of this paper is to apply the methodology for estimation and hypothesis testing to the statistical tables with nondisclosure cells. The problem of statistical tables with nondisclosure cells is one type of incomplete data problems. Then, we use the nonignorable pattern-mixture model (Little (1993a); Rubin (1987)). The pattern-mixture model requires prior information to identify the parameters of the model concerning missing data. We propose a procedure for estimation and hypothesis testing, which does not relate the distribution of nondisclosure cells to the distribution of disclosure cells. From this point of view, proposed procedure is different from any procedures based on common nonignorable pattern-mixture model, which use the model assumed for missing data mechanism. Computations for estimation and hypothesis testing are straightforward by using a direct Monte Carlo simulation method. In terms of this method, sensitivity to model assumptions can be easily assessed.

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We consider the test of equality of the latent vector and a specified vector. In this paper, we discuss three criteria for testing hypothesis. The test statistic A₁ is the inner product of the sample latent vector and the specified vector. The statistic A₂ is the α-th factor of some likelihood ratio criterion. The A₃ is the statistic given by T.W. Anderson. We calculate the percentiles based on the exact distribution of the statistic A₂. To compute the power, we obtained the non-null distribution of the statistic A₁, A₂ and A₃. And we compare the power of test using these three criteria on a bivariate and trivariate normal distribution.

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The multivariate analogue of the one sided test derived in Kudô (1963) is considered. A handy method of computing the test statistic and its significance probability is given. The method is based on applying sweep out operations on a certain matrix in a systematic manner and applying the Fortran subroutine of Sun (1988).

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In multivariate analysis, the interrelations between two sets of measurements made on the same subjects are examined by canonical correlation methods. The canonical correlation is well known as the maximum correlation between linear functions of the two sets of variables. Interest herein is to test a heterogeneity of canonical correlations of k groups based on school performance data of prenatally exposed survivors of the atomic bombing. Because the most important single factor in determining the nature and extent of the insult to the developing embryo or fetus resulting from exposure to ionizing radiation is the developmental age (gestational age), gestational ages have been grouped so as to reflect these known phases in normal development. Such a heterogeneity test statistic of k canonical correlations has been applied to the school performance data. In the findings by gestational weeks, the largest canonical correlation only was significant different from zero; but all the remaining canonical correlations were not significant. The results of multivariate analysis based on canonical and multivariate correlations disclosed effect of radiation exposure to the development of the brain in the 8-15 week and 16-25 week groups. This tendency appeared to be stronger at lower grades.

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