In this paper, we dealt with extended models which potentially improve the fit of models to data for extreme doses, as they had tolerance distributions with more flexible tails and of more flexible shape than those of the simpler two-parameter models such as the logit model. We particularly examined and compared three types of data-adaptive transformations, namely, the asymmetric power, the asymmetric odds-ratio and the symmetric power transformations. All of these models can be regarded as extentions of the logit model, which may accur as particular cases of these extended models. In our comparative studies, we used 20 data sets cited from various literatures. And further we carried out a simulation to investigate performance of these models especially with regard to the estimation of extreme doses. As a result, the asymmetric power transformation had the most flexible fit even to the skewed data.

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In this paper, we have investigated some properties and performances of the exponential power-transformation, which is appropriate for observations with negative and positive values. We have theoretically considered the distribution before the exponential power-transformation, the parameter estimation and the approximations to the estimates of the power-transforming parameter, by following the procedures in Goto et al. (1983). We have also numerically evaluated the performances of the exponential power-transformation by several examples and mid-sized simulation. The results showed that the exponential power-transformation yields a nearly symmetric distribution and it could help to regularize the data although strict normality is not achieved.

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We are going to consider a fixed significance level, exact, test of equality of three binomial proportions. At first, we carry out conditional and unconditional tests using three major test statistics and, then, compare the test sizes of them. It is observed that unconditional test is more powerful when the distribution of a test statistic is stable among conditional reference sets. However, as the discreteness of a test statistic on conditional reference set disappears, the behavior of conditional test becomes comparable to that of unconditional test. In this paper, we propose a modified statistic, which is based on the cumulative distribution of an original statistic on conditional reference set, and show that the unconditional test using the modified statistic is uniformly more powerful than the conditional test using the original statistic. Thus, we have no choice to adopt conditional test when it is possible to carry out unconditional test. We, also, propose another statistic, which is a slightly improved version of the modified statistic, and demonstrate the high performance of this statistic through numerical examples.

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The data treated in a statistical analysis become complicated, thus a lot of non-linear multivariate analysis methods have been developed. When we carry out statistical analysis, the role of computer becomes important. Although there were enough facilities for linear statistical analysis on a statistical packages or mathematics and visualization system, there were not sufficient for non-linear facilities. But recently, there are several packages with non-linear facilities. So we can carry out non-linear analysis easily. In this report we describe non-linear optimization and introduce several softwares for non-linear optimizations, non-linear statistical models and graphical representation of data, centering around free software.

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[in Japanese]

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