We are going to compare the exact unconditional powers resulting from using three well known goodness-of-fit statistics, i.e., Pearson's X², deviance and power divergence, in testing conditionally and exactly the equality of three binomial proportions. As far as I know, no paper has paid any attention to the selection of test statistics in the context of an exact conditional test. This is partly because almost all authors, apart from Mehta and Hilton (1993), have treated two binomial proportions, where signed root of each frequently used goodness-of-fit statistic is a monotonous function of an observed value on a conditional reference set. Theoretical investigations are carried out and numerical results are obtained on various settings of binomial parameters.

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In this paper, we discuss selection and ordering of variables in a discriminant analysis between two populations based on the minimization of risk. In selection of variables, under certain assumptions, we prove that the risk of discrimination being minimum for any cut-off point is a necessary and sufficient condition for the Matusita distance between two populations being maximum. In other words, under the assumptions, we “connect” a classification of an observation into one of two populations (by risk of discrimination) and separation of two populations (by Matusita distance). Then, the best selection from the full set of p variables used in the discrimination is the selection of variables that maximizes the Matusita distance between two populations. Together with this, we consider the most important variable for discrimination as the one that, when it is deleted, minimizes the Matusita distance between two populations (using p-1 variables). Under the assumptions, these methods minimize the risk of discrimination. When we have samples from two populations, we may construct appropriate hypotheses with each hypothesis expressing the Matusita distance between two populations using a particular subset of variables is maximum, and we select the best hypothesis. In order to solve the problem of multiple testing, we select the hypothesis or model (expressing the Matusita distance) that makes the best prediction or discrimination for future observations (model selection). For each hypothesis, we may obtain AIC and we select the hypothesis that minimizes the AIC. Together with this, we also investigate the ordering of variables using samples. In practice, the ordering of variables becomes important when the number of hypotheses for the selection is too large (2^p-1), while the number of hypotheses for the ordering is p (assuming large sample sizes). As an application, we have chosen the Behrens-Fisher problem. To evaluate whether the selected hypothesis maximizes the Matusita distance or the ordered hypotheses order the variables with respect to the importance of discrimination, we make Monte-Carlo simulation.

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A probability-updating method in probabilistic expert systems is considered in this paper based on minimum discrimination information. Here, newly acquired information is taken as the latest true marginal probabilities, not as observed data with the same weight as previous data. Posterior probabilities are obtained by updating prior probabilities subject to the latest true marginals. To apply this updating method to probabilistic expert systems, we extend Ku and Kullback(1968)'s minimum discrimination information method for saturated models to log-linear models, discuss localization of global updating, and show that Deming and Stephan's iterative procedure can be used to find the posterior probabilities. Our updating method can also be used to handle uncertain evidence in probabilistic expert systems.

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A new statistical hypothesis testing method is proposed for testing the equality of two curves. The overall difference between two curves is expressed as the difference of areas under the two curves. We adopt the area-difference as a test statistic whose sampling distribution is approximated by two kinds of bootstrap methods; resampling from the centered residuals and from the original paired data. Applying this method to several snow load datasets, we examine the validity of the snow load curve proposed by Matsushita and Izumi (1956) in architecture. This is done by comparing the curve with the one which is obtained by applying the least squares method to the logarithmic transformed data.

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We have developed a physical random number generator in which radioactivity, i.e., one of the most random phenomena, is used. The long-lived radioactive nuclide ²⁴¹Am and a clock pulse generator are used for generating random pulses and regular pulses, respectively. A 1024 channel scaler counts clock pulses between two consecutive random pulses. This procedure is repeated and the counts are stored in a computer. The last digit of a count at the scaler gives a digit of uniform physical random number. We have tested our random numbers for randomness and uniformity, and stored 600 million random digits on each compact disc for users.

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