arXiv subjects

Tatsuya Kubokawa

Publications and source records attributed to Tatsuya Kubokawa.

28 records · Page 2Linked to original sources

arXiv2016

Empirical Uncertain Bayes Methods in Area-level Models

Random effects model can account for the lack of fitting a regression model and increase precision of estimating area-level means. However, in case that the synthetic mean provides accurate estimates, the prior distribution may inflate an estimation error. Thus it is desirable to consider the uncertain prior distribution, which is expressed as the mixture of a one-point distribution and a proper prior distribution. In this paper, we develop an empirical Bayes approach for estimating area-level means, using the uncertain prior distribution in the context of a natural exponential family, which we call the empirical uncertain Bayes (EUB) method. The regression model considered in this paper includes the Poisson-gamma and the binomial-beta, and the normal-normal (Fay-Herriot) model, which are typically used in small area estimation. We obtain the estimators of hyperparameters based on the marginal likelihood by using a well-known Expectation-Maximization algorithm, and propose the EUB estimators of area means. For risk evaluation of the EUB estimator, we derive a second-order unbiased estimator of a conditional mean squared error by using some techniques of numerical calculation. Through simulation studies and real data applications, we evaluate a performance of the EUB estimator and compare it with the usual empirical Bayes estimator.

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arXiv2016

Consistent Estimation in Box-Cox Transformed Linear Mixed Models

The Box-Cox transformation is applied to the linear mixed models for analyzing positive and grouped data. The problem in using Box Cox transformation is that the maximum likelihood estimator of the transformation parameter is generally inconsistent. To fix it, we suggest a simple and consistent estimator for the transformation parameter based on the moment method. The consistent estimator is used to construct consistent estimators of the parameters involved in the model and under some conditions, the estimators of model parameters are shown to be consistent under $m\to\infty$, where $m$ is the number of groups. Moreover, in estimation of the expectation of the (future) observations, it is shown that the resulting estimators also hold consistency owing to the consistent estimator of the transformation parameter. The proposed estimating method is compared with the maximum likelihood method via simulation and it is shown that the proposed estimator works better than the maximum likelihood estimator. Finally, the proposed method is applied to a real data set.

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arXiv2016

Bayesian Estimators in Uncertain Nested Error Regression Models

Nested error regression models are useful tools for analysis of grouped data, especially in the case of small area estimation. This paper suggests a nested error regression model using uncertain random effects in which the random effect in each area is expressed as a mixture of a normal distribution and a positive mass at $0$. For estimation of the model parameters and prediction of the random effects, an objective Bayesian inference is proposed by setting non-informative prior distributions on the model parameters. Under mild sufficient conditions, it is shown that the posterior distribution is proper and the posterior variances are finite, confirming the validity of posterior inference. To generate samples from the posterior distribution, we provide the Gibbs sampling method with familiar forms for all the full conditional distributions. This paper also addresses the problem of predicting finite population means, and a sampling-based method is suggested to tackle this issue. Finally, the proposed model is compared with the conventional nested error regression model through simulation and empirical studies.

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arXiv2016

Heteroscedastic Nested Error Regression Models with Variance Functions

The nested error regression model is a useful tool for analyzing clustered (grouped) data, and is especially used in small area estimation. The classical nested error regression model assumes normality of random effects and error terms, and homoscedastic variances. However, these assumptions are often violated in real applications and more flexible models are required. This article proposes a nested error regression model with heteroscedastic variances, where the normality for the underlying distributions is not assumed. We propose the structure of heteroscedastic variances by using some specified variance functions and some covariates with unknown parameters. Under the setting, we construct the moment-type estimators of model parameters and some asymptotic properties including asymptotic biases and variances are derived. For predicting linear quantities including random effects, we suggest the empirical best linear unbiased predictors and the second-order unbiased estimators of mean squared errors are derived in the closed form. We investigate the proposed method with simulation and empirical studies.

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arXiv2016

Bayesian Estimators for Small Area Models Shrinking Both Means and Variances

For small area estimation of area-level data, the Fay-Herriot model is extensively used as a model based method. In the Fay-Herriot model, it is conventionally assumed that the sampling variances are known whereas estimators of sampling variances are used in practice. Thus, the settings of knowing sampling variances are unrealistic and several methods are proposed to overcome this problem. In this paper, we assume the situation where the direct estimators of the sampling variances are available as well as the sample means. Using these information, we propose a Bayesian yet objective method producing shrinkage estimation of both means and variances in the Fay-Herriot model. We consider the hierarchical structure for the sampling variances and we set uniform prior on model parameters to keep objectivity of the proposed model. For validity of the posterior inference, we show under mild conditions that the posterior distribution is proper and has finite variances. We investigate the numerical performance through simulation and empirical studies.

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arXiv2015

On Conditional Prediction Errors in Mixed Models with Application to Small Area Estimation

The empirical Bayes estimators in mixed models are useful for small area estimation in the sense of increasing precision of prediction for small area means, and one wants to know the prediction errors of the empirical Bayes estimators based on the data. This paper is concerned with conditional prediction errors in the mixed models instead of conventional unconditional prediction errors. In the mixed models based on natural exponential families with quadratic variance functions, it is shown that the difference between the conditional and unconditional prediction errors is significant under distributions far from normality. Especially for the binomial-beta mixed and the Poisson-gamma mixed models, the leading terms in the conditional prediction errors are, respectively, a quadratic concave function and an increasing function of the direct estimate in the small area, while the corresponding leading terms in the unconditional prediction errors are constants. Second-order unbiased estimators of the conditional prediction errors are also derived and their performances are examined through simulation and empirical studies.

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arXiv2015

Small Area Predictors with Dual Shrinkage of Means and Variances

The paper concerns small-area estimation in the Fay-Herriot type area-level model with random dispersions, which models the case that the sampling errors change from area to area. The resulting Bayes estimator shrinks both means and variances, but needs numerical computation to provide the estimates. In this paper, an approximated empirical Bayes (AEB) estimator with a closed form is suggested. The model parameters are estimated via the moment method, and the mean squared error of the AEB is estimated via the single parametric bootstrap method. The benchmarked estimator and a second-order unbiased estimator of the mean squared error are also derived.

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arXiv2014

Parametric Transformed Fay-Herriot Model for Small Area Estimation

In this paper, we consider parametric transformed Fay-Herriot models, and clarify conditions on transformations under which the estimator of the transformation is consistent. It is shown that the dual power transformation satisfies the conditions. Based on asymptotic properties for estimators of parameters, we derive a second-order approximation of the prediction error of the empirical best linear unbiased predictors (EBLUP) and obtain a second-order unbiased estimator of the prediction error. Finally, performances of the proposed procedures are investigated through simulation and empirical studies.

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arXiv2014

On Predictive Density Estimation for Location Families under Integrated $L_2$ and $L_1$ Losses

Our investigation concerns the estimation of predictive densities and a study of efficiency as measured by the frequentist risk of such predictive densities with integrated $L_2$ and $L_1$ losses. Our findings relate to a $p-$variate spherically symmetric observable $X \sim p_X(\|x-μ\|^2)$ and the objective of estimating the density of $Y \sim q_Y(\|y-μ\|^2)$ based on $X$. For $L_2$ loss, we describe Bayes estimation, minimum risk equivariant estimation (MRE), and minimax estimation. We focus on the risk performance of the benchmark minimum risk equivariant estimator, plug-in estimators, and plug-in type estimators with expanded scale. For the multivariate normal case, we make use of a duality result with a point estimation problem bringing into play reflected normal loss. In three of more dimensions (i.e., $p \geq 3$), we show that the MRE estimator is inadmissible under $L_2$ loss and provide dominating estimators. This brings into play Stein-type results for estimating a multivariate normal mean with a loss which is a concave and increasing function of $\|\hatμ-μ\|^2$. We also study the phenomenon of improvement on the plug-in density estimator of the form $q_Y(\|y-aX\|^2)\,, 0 1$, showing in some cases, inevitably for large enough $p$, that all choices $c>1$ are dominating estimators. Extensions are obtained for scale mixture of normals including a general inadmissibility result of the MRE estimator for $p \geq 3$. Finally, we describe and expand on analogous plug-in dominance results for spherically symmetric distributions with $p \geq 4$ under $L_1$ loss.

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arXiv2013

Dominance properties of constrained Bayes and empirical Bayes estimators

This paper studies decision theoretic properties of benchmarked estimators which are of some importance in small area estimation problems. Benchmarking is intended to improve certain aggregate properties (such as study-wide averages) when model based estimates have been applied to individual small areas. We study decision-theoretic properties of such estimators by reducing the problem to one of studying these problems in a related derived problem. For certain such problems, we show that unconstrained solutions in the original (unbenchmarked) problem give unconstrained Bayes and improved estimators which automatically satisfy the benchmark constraint. Also, dominance properties of constrained empirical Bayes estimators are shown in the Fay-Herriot model, a frequently used model in small area estimation.

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