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## A Cautionary Note on Likelihood Ratio Tests in Mixture Models

### Annals of the Institute of Statistical Mathematics (2000-09-01) 52: 481-487 , September 01, 2000

We show that iterative methods for maximizing the likelihood in a mixture of exponentials model depend strongly on their particular implementation. Different starting strategies and stopping rules yield completely different estimators of the parameters. This is demonstrated for the likelihood ratio test of homogeneity against two-component exponential mixtures, when the test statistic is calculated by the EM algorithm.

## Report of the survey design for agricultural production estimates in the Ryukyu Islands

### Annals of the Institute of Statistical Mathematics (1951-12-01) 3: 109-121 , December 01, 1951

## A new instrumental variable estimation for diffusion processes

### Annals of the Institute of Statistical Mathematics (2005-12-01) 57: 733-745 , December 01, 2005

We consider the problem of parametric inference from continuous sample paths of the diffusion processes {*x(t)}* generated by the system of possibly nonstationary and/or nonlinear Ito stochastic differential equations. We propose a new instrumental variable estimator of the parameter whose pivotal statistic has a Gaussian distribution for all possible values of parameter. The new estimator enables us to construct exact level-α confidence intervals and tests for the parameter in the possibly non-stationary and/or nonlinear diffusion processes. Applications to several non-stationary and/or nonlinear diffusion processes are considered as examples.

## Robust estimation of generalized partially linear model for longitudinal data with dropouts

### Annals of the Institute of Statistical Mathematics (2016-10-01) 68: 977-1000 , October 01, 2016

In this paper, we study the robust estimation of generalized partially linear models (GPLMs) for longitudinal data with dropouts. We aim at achieving robustness against outliers. To this end, a weighted likelihood method is first proposed to obtain the robust estimation of the parameters involved in the dropout model for describing the missing process. Then, a robust inverse probability-weighted generalized estimating equation is developed to achieve robust estimation of the mean model. To approximate the nonparametric function in the GPLM, a regression spline smoothing method is adopted which can linearize the nonparametric function such that statistical inference can be conducted operationally as if a generalized linear model was used. The asymptotic properties of the proposed estimator are established under some regularity conditions, and simulation studies show the robustness of the proposed estimator. In the end, the proposed method is applied to analyze a real data set.

## A generalized partially linear framework for variance functions

### Annals of the Institute of Statistical Mathematics (2017-10-04): 1-29 , October 04, 2017

When model the heteroscedasticity in a broad class of partially linear models, we allow the variance function to be a partial linear model as well and the parameters in the variance function to be different from those in the mean function. We develop a two-step estimation procedure, where in the first step some initial estimates of the parameters in both the mean and variance functions are obtained and then in the second step the estimates are updated using the weights calculated based on the initial estimates. The resulting weighted estimators of the linear coefficients in both the mean and variance functions are shown to be asymptotically normal, more efficient than the initial un-weighted estimators, and most efficient in the sense of semiparametric efficiency for some special cases. Simulation experiments are conducted to examine the numerical performance of the proposed procedure, which is also applied to data from an air pollution study in Mexico City.

## Joint Distribution of Rises and Falls

### Annals of the Institute of Statistical Mathematics (2000-09-01) 52: 415-425 , September 01, 2000

The marginal distributions of the number of rises and the number of falls have been used successfully in various areas of statistics, especially in non-parametric statistical inference. Carlitz (1972, *Duke Math. J.**39*, 268–269) showed that the generating function of the joint distribution for the numbers of rises and falls satisfies certain complex combinatorial equations, and pointed out that he had been unable to derive the explicit formula for the joint distribution from these equations. After more than two decades, this latter problem remains unsolved. In this article, the joint distribution is obtained via the probabilistic method of finite Markov chain imbedding for random permutations. A numerical example is provided to illustrate the theoretical results and the corresponding computational procedures.

## 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

## Bayesian calibration in the estimation of the age of rhinoceros

### Annals of the Institute of Statistical Mathematics (1996-03-01) 48: 17-28 , March 01, 1996

In this paper the Bayesian approach for nonlinear multivariate calibration will be illustrated. This goal will be achieved by applying the Gibbs sampler to the rhinoceros data given by Clarke (1992, *Biometrics*, *48*(4), 1081–1094). It will be shown that the point estimates obtained from the profile likelihoods and those calculated from the marginal posterior densities using improper priors will in most cases be similar.

## Minimum Divergence Estimators Based on Grouped Data

### Annals of the Institute of Statistical Mathematics (2001-06-01) 53: 277-288 , June 01, 2001

The paper considers statistical models with real-valued observations i.i.d. by *F*(*x*, θ_{0}) from a family of distribution functions (*F*(*x*, θ); θ ε Θ), Θ ⊂ *R*^{s}, *s* ≥ 1. For random quantizations defined by sample quantiles (*F*_{n}^{−1} (λ_{1}),θ, *F*_{n}^{−1} (λ_{m−1})) of arbitrary fixed orders 0 < λ_{1} θ < λ_{m-1} < 1, there are studied estimators θ_{φ,n} of θ_{0} which minimize φ-divergences of the theoretical and empirical probabilities. Under an appropriate regularity, all these estimators are shown to be as efficient (first order, in the sense of Rao) as the MLE in the model quantified nonrandomly by (*F*^{−1} (λ_{1},θ_{0}),θ, *F*^{−1} (λ_{m−1}, θ_{0})). Moreover, the Fisher information matrix *I*_{m} (θ_{0}, λ) of the latter model with the equidistant orders λ = (λ_{j} = *j*/*m* : 1 ≤ *j* ≤ *m* − 1) arbitrarily closely approximates the Fisher information *J*(θ_{0}) of the original model when *m* is appropriately large. Thus the random binning by a large number of quantiles of equidistant orders leads to appropriate estimates of the above considered type.