A random loading model is presented as a numerical model in a Monte Carlo experiment of factor analysis. Three methods for the number-of-factors problem, Guttman-Kaiser, likelihood ratio and AIC, and two least-squares estimators, one-step and final, were compared experimentally within each category. On the whole, AIC method using the final estimator fared best, whereas the one-step estimator behaved better than the final estimator in estimating the unique variances.

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Recursive procedures proposed in this paper can find the maximum likelihood estimates (MLEs) for hierarchical loglinear models more efficiently than the iterative proportional fitting procedure (IPFP), the expectation-maximization(EM) algorithm and the Newton-Raphson method, especially for higher dimensional contingency tables. For a given loglinear model, at first, the recursive procedures separate it recursively into a class of models of marginal tables with the lowest possible dimensions, secondly find the MLEs for the respective lower dimensional models, and finally the proposed procedures obtain the MLEs for the original higher dimensional model from the MLEs of these lower dimensional models. For the lower dimensional models unable to be separated further, the recursive procedures find the MLEs by using the IPFP, the EM algorithm or the Newton-Raphson method.

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A method of variable selection is introduced in Hayashi's third method of quantification. Two numerical examples of geological applications are given to show its performance.

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As a powerful method to predict an objective variable from a set of k explanatory variables, we have multiple regression analysis using polynomial models. However, for the data having non-linear structure, this method has problems such as difficulty in the determination of the degree of polynomial, excessive large number of parameters to be estimated. So, in many cases, we assume a linear or quadratic polynomial model and try to fit a k-dimensional hyperplane or hypersurface to the given data in the (k+1)-dimensional real space R^k+1, independently of the structure of the data. For this reason, the accuracy of prediction is poor, particularly for the data having non-linear structure.
In the present paper, we propose a few models in which we take the structure of data into consideration in order to obtain estimates with higher accuracy. The proposed models perform prediction by transforming the points in R^k which are the values of the k explanatory variables to a 2-dimensional plane R² and fitting a regression plane on R², based on the information of higher moments of the explanatory variables.

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A Basic Oxygen Furnace(BOF), which refines steel, is a typical batch type process, and it has been computer-cotrolled with mathematical models. Since the conditions of materials for the process often vary from time to time, it is important to periodically update the models to keep up with the change of the process.
In the Mizushima works, a model analysis system for BOF refining cotrol has been developed in order to easily maintain the accuracy of the models. The system consists of four parts; data extraction, simulation, accuracy evaluation and statistical analysis, thereby statistical models such as multi-regression equations can be automatically analyzed, using accumulated actual data.
The system has contributed to the improvement of the model accuracy and consequently to the saving of sub-materials and better hit rate of refining.

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In this paper we give a distribution of the Hilbert modular type cusp forms over real quadratic number fields, which are closely connected with the Euler number of the Hilbert modular surface, using the trace formula proved by the a uthor. We consider the statistical model for the increasing trend of the dimensions, and analyze their distribution through it.

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Let X and Y be sample means drawn from two independent normal distributions with unknown mean μ_i and known variance σ²_i for i=1, 2, respectively, and let μ*≡max(μ₁, μ₂). The purpose of the present paper is to propose a new estimator of μ* as a hybrid estimator η(C) with a given non-negative C, which includes both estimators, an estimator η₁ of a linear combination of X and Y and a simple estimator η₂=max(X, Y). After comparing η(C) numerically with the maximum likelihood estimator δ given by Blumenthal and Cohen (1968b), it is found that η(C) is better than δ for C∈[1.87, 1.91] in view of the mean square error. The C has also been investigated in the sense of minimax regret, and an optimal C is obtained numerically as C^*=1.704.

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A defect inherent in the traditional goodness-of-fit χ²-test is pointed out, see Miyatake (1985), in which the number of classes cannot be fixed a priori. Let a population be divided into l classes and χ²_l-1 be the χ²-value obtained from a hypothesis H₀ and observed frequency in each class. We assume that l is even, and put together these classes two by two, getting a coarse division of k classes (k=l/2). Let χ²_k-1 be the χ²-value obtained from this coarse division under the same H₀ and observed frequencies. Then we get the decomposition χ²_l-1=χ²_k-1+χ²_l-k, where χ²_l-k is a χ²-variable of degree of freedom l-k. It will be shown that the use of the pair (χ²_k-1, χ²_l-k) is more effective in the goodness-of-fit test, in comparison with the traditional method using the single χ²_l-1.

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3SLS is more efficient than 2SLS, if at least one equation in a simultaneous equation model is over-identifiable. It is difficult to build in one computer-run a model which is not only econometrically satisfactory but also reasonable for economic research. We define such a model and propose a computer-assisted method to obtain it efficiently. Concretely speaking, we formulate the j-th best subset problem for (C)2SLS, propose a method to find the first 3 or 4 best equation candidates, which are the solutions to the previous problems, formulate the best model problem for (C)3SLS, and suggest that the model be selected which passes the conditions and criteria for sign and magnitude tests, statistical tests, and data-analytic tests, and then has the highest adjusted total coefficient of determination. Various programming techniques are proposed. The System OEPP can be used for the proposed method.

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A FORTRAN program using Simpson's rule is reported for computing exact p-values associated with Tukey's and Dunnett's multiple comparisons procedures in an imbalanced one-way ANOVA model. A FORTRAN program for Dunnett's test is provided by Dunlap, Marx and Agamy (1981) in the case of all the sample sizes of treatment groups, except for a control group, being homogeneous. We modify their program to keep better computational accuracy and extend it to imbalanced Tukey's and Dunnett's tests. We investigate the computational accuracy and CPU times by applying it to many actual critical values in some published tables. Exact p-values of Tukey's test for some critical values of Hunter method, which are given by Stoline (1981) as the examples that Tukey-Kramer method is slightly more conservative than Hunter method for certain imbalanced cases, are illustrated with the corresponding approximate p-values of Tukey-Kramer's test.

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