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## On the prediction of phenomena from qualitative data and the quantification of qualitative data from the mathematico-statistical point of view

### Annals of the Institute of Statistical Mathematics (1951-12-01) 3: 69-98 , December 01, 1951

## Proof of the law of iterated logarithm through diffusion equation

### Annals of the Institute of Statistical Mathematics (1959-03-01) 10: 21-28 , March 01, 1959

## Bayesian detection of structural changes

### Annals of the Institute of Statistical Mathematics (1991-03-01) 43: 77-93 , March 01, 1991

A Bayesian solution is given to the problem of making inferences about an unknown number of structural changes in a sequence of observations. Inferences are based on the posterior distribution of the number of change points and on the posterior probabilities of possible change points. Detailed analyses are given for binomial data and some regression problems, and numerical illustrations are provided. In addition, an approximation procedure to compute the posterior probabilities is presented.

## Asymptotic behavior of M-estimator and related random field for diffusion process

### Annals of the Institute of Statistical Mathematics (1990-06-01) 42: 221-251 , June 01, 1990

The *M*-estimate which maximizes a positive stochastic process *Q* is treated for multidimensional diffusion models. The convergence in distribution of the process of ratio of *Q*'s after normalizing is proved. The asymptotic behavior of *M*-estimates is stated. We present the asymptotic variance in general cases and in estimation by misspecified models.

## Extreme sizes in Gibbs-type exchangeable random partitions

### Annals of the Institute of Statistical Mathematics (2017-02-01) 69: 1-37 , February 01, 2017

Gibbs-type exchangeable random partition, which is a class of multiplicative measures on the set of positive integer partitions, appear in various contexts, including Bayesian statistics, random combinatorial structures, and stochastic models of diversity in various phenomena. Some distributional results on ordered sizes in the Gibbs partition are established by introducing associated partial Bell polynomials and analysis of the generating functions. The combinatorial approach is applied to derive explicit results on asymptotic behavior of the extreme sizes in the Gibbs partition. Especially, Ewens–Pitman partition, which is the sample from the Poisson–Dirichlet process and has been discussed from rather model-specific viewpoints, and a random partition which was recently introduced by Gnedin, are discussed in the details. As by-products, some formulas for the associated partial Bell polynomials are presented.

## The significance of the discordant variance estimates

### Annals of the Institute of Statistical Mathematics (1955-12-01) 7: 39-55 , December 01, 1955

## A note on the use of median ranges

### Annals of the Institute of Statistical Mathematics (1962-12-01) 14: 87-89 , December 01, 1962

## Quasi-Periodic Process Modeling

### Smoothness Priors Analysis of Time Series (1996-01-01) 116: 189-200 , January 01, 1996

The early development of time series has been related to a quest for an understanding of cyclical phenomena. For example, Shuster’s periodogram (1898, 1906), and Yule’s (1927) introduction of autoregressive models, were devoted to the analysis of cyclical sunspot numbers and Whittle’s (1954) analysis of the water level in a rock channel on the Wellington coast of New Zealand is also related to a cyclical phenomenon. In fact many time series exhibit cyclical behavior in the sense there appears to be an approximate repetition of a pattern with a not very well defined period or amplitude. Rather both the period and amplitude appear to change gradually. Typically ecological data, air pollution data and several other physical phenomena exhibit such behavior. Two of the most familiar examples of time series which exhibit such behavior and which have been extensively analyzed are the Canadian lynx data, (for example see Campbell and Walker 1977, Tong 1977, Bhansali 1979 and Priestly 1981), and the Wolfer sunspot series, (Morris 1977, Tong 1983). Such series are frequently modeled by AR, ARMA or AR plus sinusoidal models. However, none of these modeling methods are very satisfactory for the prediction of more than one lead time, (Tong 1982). As pointed out in Akaike (1977b) in his discussion of several analyses of the Canadian lynx data, those analyses were unconvincing and that the critical issue in modeling time series is the selection of a proper model.

## Randomized unbiased estimation of restricted parameters

### Annals of the Institute of Statistical Mathematics (1963-12-01) 15: 61-66 , December 01, 1963

## On a min-max theorem and some of its applications

### Annals of the Institute of Statistical Mathematics (1960-02-01) 12: 1-5 , February 01, 1960

### Summary

A theorem is obtained which enables us to get the arrangement which gives the minimum of the maximum diagonal element in all the possible rearrangements of a matrix of which elements are arranged in the decreasing order of magnitude in each row.

Some implications of the theorem with the rational economic behavior of consumers and with the preparation for the Seidel's process of successive approximation are discussed.