Four variable selection procedures with low computaional cost are proposed in principal component analysis based on a subset of variables (Tanaka & Mori, 1997), which select a reasonable subset of variables representing all the variables very well. They are Backward elimination, Forward selection, Backward-forward stepwise selection and Forward-backward stepwise selection. Through some numerical examples, their performances are evaluated in comparison with the results of all possible selection procedure and previous methods of variable selection in principal component analysis.

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In a two-sample model, we introduce the normal theory statistical procedures, non-parametric procedures based on ranks and the semi-parametric procedures possessing the robustness of Huber (1964). After the computational algorithm of the permutation tests for their test procedures is given, the powers of the three test procedures are compared due to a simulation. The computational algorithm for the semi-parametric estimators is also given. The mean squared errors for the three-type estimators are compared through a simulation. It can be seen that, (i) when the underlying distribution is in a neighbourhood of the normal distribution, the semi-parametric procedures is superior to the parametric procedures and the non-parametric procedures, and (ii) when the underlying distribution is not in a neighbourhood of the normal distribution, the non-parametric procedures are superior. By using the bootstrap estimate of the variances for the tree-type estimators based on real data, we may give the best choice among the three statistical procedures. However we point out that the bootstrap choice is sometimes mistaken. Furthermore, by using a simulation of 1000 replicates, we can see that the bootstrap estimate of the relative effiency among the estimators is fairly different from the value of the real relative efficiency.

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The aim of the present study was to analyze some problems on the education for the hearing impaired by the method of the computational statistics. The problem here is if there were any differences between the hearing impaired students and the normal hearing staff in the evaluation of essays. 8 essays were written by 4 hearing impaired students of Tsukuba College of Technology and 4 normal hearing students of other universities. 65 evaluators were selected on the condition of the hearing ability and the age group. Thus, the data set including the evaluated results was gotten as 65×8 matrix. It was analyzed by constrained PCA (CPCA). The statistical analysis resulted in 4 useful hints. The procedure for getting a useful hint is as follows. At first, an interpretation is guessed by the coefficients of CPCA. Then, the interpretation takes a verification test on the original dataset. After these processes, it is fixed as a useful hint. Now, those statistical analysis activities are continuing for the education of the hearing impaired. The computer program of CPCA written in S-PLUS is shown.

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We introduce statistical tests for pseudorandom number generations, which are based on functionals of sample paths of random walks. We consider the following pseudorandom number generatios : m-sequences, additive number generators, cellular automata generators. The results of random walk tests detect statistical biases of these generators. Moreover, from the results we can find some conjectures, for example, ● The multiples of a trinomial f over GF (2) are only those having the form such as f^2, f^4, and so on, while the degrees of multiples are not so large. This conjecture has been proved by Munemasa (1998) in case that the degree of multiples are less than or equal to the twice of the degree of f. ● A primitive polynomial over GF (2) and ite reciprocal polynomial must have different algebraic properties. This conjecture is assured by results of maximum tests, and sojourn time tests.

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[in Japanese]

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