Anonymized Data are defined so that no individual shall be identified. This unidentifiability, however, is not clearly defined. Hence the assessment process of this unidentifiability has not been clearly formulated, which results in few consistent arguments on the improvement of the process. Therefore the present paper substantiates one clear method to decide whether given data are identifiable or not by measuring re-identification risk. The existing theory of re-identification risk lacks the method of deciding its critical value; the present paper statistically estimates it using a fact that identification has not been observed. Our evidence based method, supported by facts, ensures lasting improvements on the institution of Anonymized Data. The present paper actually proposes concrete improvements on its assessment process.

In the high-dimensional analysis, usual estimation procedures yield large estimation errors because sample sizes are small as compared with the number of unknown parameters. Also in the small-area estimation, the sample means of small-areas have large estimation errors since the sample sizes from individual areas are small. In this case, shrinkage estimators are known as stable procedures with higher precision, because the shrinkage estimators can be obtained by shrinking the usual estimators toward stable estimators which are derived under some reasonable assumptions. In this paper, usefulness of shrinkage procedures is explained from applied aspects in high-dimensional analysis and small area estimation.

This paper proposes constructing the confidence sets for the break date in linear regression models with non-homogeneous regressors. We first derive optimal tests for the location of the break date and then construct the confidence sets by inverting the tests. It turns out that, in general, the asymptotic distributions of the test statistics for the location of the break date depend on nuisance parameters, the moments of the non-homogeneous regressors, but we propose the solution to this problem when a linear trend is only the non-homogeneous regressor. The method proposed in the paper works well in finite samples in that the empirical coverage rates are close to the nominal confidence level and thus our method is practically useful.

The wrapped Cauchy distribution or the circular Cauchy distribution is a probability distribution defined on the circle. In this paper we discuss two topics related to the wrapped Cauchy distribution. First some known results about this distribution are introduced, including basic properties, derivations, parameter estimation, association with the Möbius transformation, and comparison with the von Mises distribution. Second we provide a review of statistical models related to the wrapped Cauchy. In particular, the most of the second topic is devoted to introduce a bivariate extension of the wrapped Cauchy distribution proposed by Kato and Pewsey (2015). It is seen that this distribution has numerous tractable properties in terms of probability density function, interpretation of the parameters, marginals and conditionals, correlation coefficients, and parameter estimation.