This paper presents a moving average of independent random variables with normal distributions that approximates a stochastic process whose sample paths are periodic (we call it the periodic stochastic process). Since the periodic stochastic process does not have a spectral density, it can not be directly represented as a moving average according to the Wold decomposition theorem. The results of this paper are twofold. First, we point out that the theorem originally proved by Slutzky (1937) is not satisfactory in the sense that the moving average process constructed by him does not converge to any processes in L 2 as the sum of white noise goes to infinity though the spectral distribution of it weakly converges to a step function which is the spectral distribution of a periodic stochastic process. Secondly we propose a new moving average process that approximates a nontrivial periodic stochastic process in L 2 and almost surely.
{"title":"AN APPROXIMATE MOVING AVERAGE REPRESENTATION OF THE PERIODIC STOCHASTIC PROCESS","authors":"H. Kato","doi":"10.14490/JJSS.43.1","DOIUrl":"https://doi.org/10.14490/JJSS.43.1","url":null,"abstract":"This paper presents a moving average of independent random variables with normal distributions that approximates a stochastic process whose sample paths are periodic (we call it the periodic stochastic process). Since the periodic stochastic process does not have a spectral density, it can not be directly represented as a moving average according to the Wold decomposition theorem. The results of this paper are twofold. First, we point out that the theorem originally proved by Slutzky (1937) is not satisfactory in the sense that the moving average process constructed by him does not converge to any processes in L 2 as the sum of white noise goes to infinity though the spectral distribution of it weakly converges to a step function which is the spectral distribution of a periodic stochastic process. Secondly we propose a new moving average process that approximates a nontrivial periodic stochastic process in L 2 and almost surely.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"368 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132577247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Two Bayesian prediction problems in the context of model averaging are investigated by adopting dual Kullback-Leibler divergence losses, the e-divergence and the m-divergence losses. We show that the optimal predictors under the two losses are shown to satisfy interesting saddlepoint-type equalities. Actually, the optimal predictor under the e-divergence loss balances the log-likelihood ratio and the loss, while the optimal predictor under the m-divergence loss balances the Shannon entropy difference and the loss. These equalities also hold for the predictors maximizing the log-likelihood and the Shannon entropy respectively under the e-divergence loss and the m-divergence loss, showing that enlarging the log-likelihood and the Shannon entropy moderately will lead to the optimal predictors. In each divergence loss case we derive a robust predictor in the sense that its posterior risk is constant by minimizing a certain convex function. The Legendre transformation induced by this convex function implies that there is inherent duality in each Bayesian prediction problem.
{"title":"Twofold structure of duality in Bayesian model averaging","authors":"Toshio Ohnishi, T. Yanagimoto","doi":"10.14490/JJSS.43.29","DOIUrl":"https://doi.org/10.14490/JJSS.43.29","url":null,"abstract":"Two Bayesian prediction problems in the context of model averaging are investigated by adopting dual Kullback-Leibler divergence losses, the e-divergence and the m-divergence losses. We show that the optimal predictors under the two losses are shown to satisfy interesting saddlepoint-type equalities. Actually, the optimal predictor under the e-divergence loss balances the log-likelihood ratio and the loss, while the optimal predictor under the m-divergence loss balances the Shannon entropy difference and the loss. These equalities also hold for the predictors maximizing the log-likelihood and the Shannon entropy respectively under the e-divergence loss and the m-divergence loss, showing that enlarging the log-likelihood and the Shannon entropy moderately will lead to the optimal predictors. In each divergence loss case we derive a robust predictor in the sense that its posterior risk is constant by minimizing a certain convex function. The Legendre transformation induced by this convex function implies that there is inherent duality in each Bayesian prediction problem.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131837515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Masakazu Fujiwara, T. Minamidani, Isamu Nagai, H. Wakaki
Principal components analysis (PCA) is one method for reducing the dimension of the explanatory variables, although the principal components are derived by using all the explanatory variables. Several authors have proposed a modified PCA (MPCA), which is based on using only selected explanatory variables in order to obtain the principal components (see e.g., Jolliffie, 1972, 1986; Robert and Escoufier, 1976; Tanaka and Mori, 1997). However, MPCA uses all of the selected explanatory variables to obtain the principal components. There may, therefore, be extra variables for some of the principal components. Hence, in the present paper, we propose a generalized PCA (GPCA) by extending the partitioning of the explanatory variables. In this paper, we estimate the unknown vector in the linear regression model based on the result of a GPCA. We also propose some improvements in the method to reduce the computational cost.
主成分分析(PCA)是一种降低解释变量维数的方法，尽管主成分是由所有解释变量导出的。几位作者提出了一种改进的PCA (MPCA)，它基于仅使用选定的解释变量来获得主成分(例如，Jolliffie, 1972, 1986;Robert and Escoufier, 1976;田中和森，1997)。然而，MPCA使用所有选定的解释变量来获得主成分。因此，对于某些主成分，可能存在额外的变量。因此，在本文中，我们通过扩展解释变量的划分，提出了广义主成分分析(GPCA)。在本文中，我们基于GPCA的结果估计了线性回归模型中的未知向量。我们还提出了一些改进方法，以减少计算成本。
{"title":"Principal Components Regression by using Generalized Principal Components Analysis","authors":"Masakazu Fujiwara, T. Minamidani, Isamu Nagai, H. Wakaki","doi":"10.14490/JJSS.43.57","DOIUrl":"https://doi.org/10.14490/JJSS.43.57","url":null,"abstract":"Principal components analysis (PCA) is one method for reducing the dimension of the explanatory variables, although the principal components are derived by using all the explanatory variables. Several authors have proposed a modified PCA (MPCA), which is based on using only selected explanatory variables in order to obtain the principal components (see e.g., Jolliffie, 1972, 1986; Robert and Escoufier, 1976; Tanaka and Mori, 1997). However, MPCA uses all of the selected explanatory variables to obtain the principal components. There may, therefore, be extra variables for some of the principal components. Hence, in the present paper, we propose a generalized PCA (GPCA) by extending the partitioning of the explanatory variables. In this paper, we estimate the unknown vector in the linear regression model based on the result of a GPCA. We also propose some improvements in the method to reduce the computational cost.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121129453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the linear regression model with a spatially correlated error term on a lattice process. When we estimate coefficients in the linear regression model, the generalized least squares estimator (GLSE) is used if the covariance structures are known. However, the GLSE for large spatial data sets is impractically time-consuming because it includes the inversion of the covariance matrix of error terms in different spatial points that is the size of the number of observations. To reduce the computational complexity, we propose the pseudo best estimator (PBE) using spatial covariance structures approximated by separable covariance functions. We derive the asymptotic covariance matrix of the PBE and compare it with those of the least squares estimator (LSE) and the GLSE through some simulations. They also imply that the effect of the misspecification of the covariance matrix for the GLSE is examined. Monte Carlo simulations demonstrate the improvement of the LSE, which does not contain the information of the spatial covariance structure, by the PBE using separable covariance functions even if the true process has an isotropic Matern covariance function. Additionally our proposed PBE is computationally efficient relative to the GLSE for large spatial data sets.
{"title":"Pseudo Best Estimator by a Separable Approximation of Spatial Covariance Structures","authors":"Toshihiro Hirano","doi":"10.14490/JJSS.44.43","DOIUrl":"https://doi.org/10.14490/JJSS.44.43","url":null,"abstract":"We consider the linear regression model with a spatially correlated error term on a lattice process. When we estimate coefficients in the linear regression model, the generalized least squares estimator (GLSE) is used if the covariance structures are known. However, the GLSE for large spatial data sets is impractically time-consuming because it includes the inversion of the covariance matrix of error terms in different spatial points that is the size of the number of observations. To reduce the computational complexity, we propose the pseudo best estimator (PBE) using spatial covariance structures approximated by separable covariance functions. We derive the asymptotic covariance matrix of the PBE and compare it with those of the least squares estimator (LSE) and the GLSE through some simulations. They also imply that the effect of the misspecification of the covariance matrix for the GLSE is examined. Monte Carlo simulations demonstrate the improvement of the LSE, which does not contain the information of the spatial covariance structure, by the PBE using separable covariance functions even if the true process has an isotropic Matern covariance function. Additionally our proposed PBE is computationally efficient relative to the GLSE for large spatial data sets.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130521639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider two normal populations Π1 and Π2 with means μ1 and μ2 and variances σ2 1 and σ 2 2, respectively, where μ1, μ2, σ 2 1, and σ 2 2 are unknown. Having observed X11, . . . , X1r from Π1 and X21, . . . , X2s from Π2, it is required to estimate μ1 − μ2 by X̄1(r) − X̄2(s) within ±d, where X̄1(r) = 1r ∑r i=1 X1i, X̄2(s) = 1 s ∑s i=1 X2i, and d(> 0) is a given constant. In order to meet such a requirement, we construct a confidence interval
{"title":"Robustness of a Two-Stage Estimation Procedure when Variances are Unequal","authors":"Y. Takada, Katuya Miyama","doi":"10.14490/JJSS.42.207","DOIUrl":"https://doi.org/10.14490/JJSS.42.207","url":null,"abstract":"We consider two normal populations Π1 and Π2 with means μ1 and μ2 and variances σ2 1 and σ 2 2, respectively, where μ1, μ2, σ 2 1, and σ 2 2 are unknown. Having observed X11, . . . , X1r from Π1 and X21, . . . , X2s from Π2, it is required to estimate μ1 − μ2 by X̄1(r) − X̄2(s) within ±d, where X̄1(r) = 1r ∑r i=1 X1i, X̄2(s) = 1 s ∑s i=1 X2i, and d(> 0) is a given constant. In order to meet such a requirement, we construct a confidence interval","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117144178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The effect of trawling on seabed fauna in the Northern Prawn Fishery experimental region of Australia is investigated through distributional changes in individual weights for each species. A stochastic growth model is employed to overcome a limited number of effective observations. One statistical challenge is to deal with non-identically distributed observations as only total weights and numbers of indi
{"title":"Detection of Ecological Disturbances to Seabed Fauna through Change of Weight Distribution","authors":"Mayumi Naka, R. Shibata, R. Darnell","doi":"10.14490/JJSS.42.185","DOIUrl":"https://doi.org/10.14490/JJSS.42.185","url":null,"abstract":"The effect of trawling on seabed fauna in the Northern Prawn Fishery experimental region of Australia is investigated through distributional changes in individual weights for each species. A stochastic growth model is employed to overcome a limited number of effective observations. One statistical challenge is to deal with non-identically distributed observations as only total weights and numbers of indi","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123043029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a family of robust nonparametric estimators for regression function based on the kernel method. We establish the almost complete convergence rate of these estimators under the α-mixing assumption and on the concentration properties on small balls of the probability measure of the functional regressors. Some applications to physics real data have been made. These results are extensions to dependent data of the results given by Azzedine et al. (2008).
{"title":"ROBUST REGRESSION FOR FUNCTIONAL TIME SERIES DATA","authors":"M. Attouch, Ali Laksaci, E. O. Saïd","doi":"10.14490/JJSS.42.125","DOIUrl":"https://doi.org/10.14490/JJSS.42.125","url":null,"abstract":"We propose a family of robust nonparametric estimators for regression function based on the kernel method. We establish the almost complete convergence rate of these estimators under the α-mixing assumption and on the concentration properties on small balls of the probability measure of the functional regressors. Some applications to physics real data have been made. These results are extensions to dependent data of the results given by Azzedine et al. (2008).","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121913790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper develops an asymptotic expansion of a percentile point of the Ginibased standardized sample mean. Such approximate percentiles can be used for proposing tests of hypotheses or confidence intervals of μ when samples arrive from a normal distribution with unknown mean μ and standard deviation σ. We have asymptotically expressed the percentile point bm,α of the Gini-based pivot (1.5), that is, the Gini-based standardized sample mean. Using large-scale simulations, approximations, and data analyses, we report that the Gini-based test and confidence interval procedures for μ perform better or practically as well as the customarily employed Student’s t-based procedures when samples arrive from a normal distribution with suspect outliers. This interesting finding is especially noteworthy when we have a small random sample from a normal population with possible outliers.
{"title":"ASYMPTOTIC EXPANSION OF THE PERCENTILES FOR A SAMPLE MEAN STANDARDIZED BY GMD IN A NORMAL CASE WITH APPLICATIONS","authors":"N. Mukhopadhyay, Bhargab Chattopadhyay","doi":"10.14490/JJSS.42.165","DOIUrl":"https://doi.org/10.14490/JJSS.42.165","url":null,"abstract":"This paper develops an asymptotic expansion of a percentile point of the Ginibased standardized sample mean. Such approximate percentiles can be used for proposing tests of hypotheses or confidence intervals of μ when samples arrive from a normal distribution with unknown mean μ and standard deviation σ. We have asymptotically expressed the percentile point bm,α of the Gini-based pivot (1.5), that is, the Gini-based standardized sample mean. Using large-scale simulations, approximations, and data analyses, we report that the Gini-based test and confidence interval procedures for μ perform better or practically as well as the customarily employed Student’s t-based procedures when samples arrive from a normal distribution with suspect outliers. This interesting finding is especially noteworthy when we have a small random sample from a normal population with possible outliers.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125134570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider a parallel profile model for several groups. Given the parallel profile model we construct tests based on the likelihood ratio, without any restrictions on the parameter ...
{"title":"Profile Analysis with Random-Effects Covariance Structure","authors":"M. Srivastava, M. Singull","doi":"10.14490/JJSS.42.145","DOIUrl":"https://doi.org/10.14490/JJSS.42.145","url":null,"abstract":"In this paper, we consider a parallel profile model for several groups. Given the parallel profile model we construct tests based on the likelihood ratio, without any restrictions on the parameter ...","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132649970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is a common practice to analyze complex longitudinal data using flexible nonlinear mixed-effects (NLME) models with normality assumption. However, a serious departure of normality may cause lack of robustness and subsequently lead to invalid inference and unreasonable estimates. Covariates are usually introduced in such models to partially explain inter-subject variations, but some covariates may be often measured with substantial errors. Moreover, the response observations may be subject to left-censoring due to a detection limit. Inferential procedures can be complicated dramatically when data with asymmetric (skewed) characteristics, leftcensoring and measurement errors are observed. In the literature, there has been considerable interest in accommodating either skewness, censoring or covariate measurement errors in such models, but there is relatively little work concerning all of the three features simultaneously. In this article, we jointly investigate a skew-t NLME model for response (with left-censoring) process and a skew-t nonparametric mixedeffects model for covariate (with measurement errors) process. We propose a robust skew-t Bayesian modeling approach in a general form to analyze data in capturing the effects of skewness, censoring and measurement errors in covariates simultaneously. A real data example is offered to illustrate the methodologies. The proposed modeling alternative offers important advantages in the sense that the model can be easily fitted in freely available software and the computational effort for the model with a skew-t distribution is almost equivalent to that of the model with a standard normal distribution.
{"title":"Simultaneous Bayesian Inference for Longitudinal Data with Asymmetry, Left-censoring and Covariates Measured with Errors","authors":"Yangxin Huang, G. Dagne","doi":"10.14490/JJSS.42.1","DOIUrl":"https://doi.org/10.14490/JJSS.42.1","url":null,"abstract":"It is a common practice to analyze complex longitudinal data using flexible nonlinear mixed-effects (NLME) models with normality assumption. However, a serious departure of normality may cause lack of robustness and subsequently lead to invalid inference and unreasonable estimates. Covariates are usually introduced in such models to partially explain inter-subject variations, but some covariates may be often measured with substantial errors. Moreover, the response observations may be subject to left-censoring due to a detection limit. Inferential procedures can be complicated dramatically when data with asymmetric (skewed) characteristics, leftcensoring and measurement errors are observed. In the literature, there has been considerable interest in accommodating either skewness, censoring or covariate measurement errors in such models, but there is relatively little work concerning all of the three features simultaneously. In this article, we jointly investigate a skew-t NLME model for response (with left-censoring) process and a skew-t nonparametric mixedeffects model for covariate (with measurement errors) process. We propose a robust skew-t Bayesian modeling approach in a general form to analyze data in capturing the effects of skewness, censoring and measurement errors in covariates simultaneously. A real data example is offered to illustrate the methodologies. The proposed modeling alternative offers important advantages in the sense that the model can be easily fitted in freely available software and the computational effort for the model with a skew-t distribution is almost equivalent to that of the model with a standard normal distribution.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127777801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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