The weighted entropy introduced by Belis and Guiasu (1968) is viewed as a measure of uncertainty. Di Crescenzo and Longobardi (2006) proposed dynamic form of these measure namely weighted residual (WRE) and past entropies (WPE). In this paper, we extend the definition of weighted residual and past entropies to bivariate setup and obtain some of its properties. Several properties, including monotonicity and bounds of BWRE and BWRP are obtained. We also look into the problem of extending WRE and WPE for conditionally specified models. Several properties, including bounds of CWRE and CWPE are obtained for conditional distributions. It is shown that the proposed measure uniquely determines the distribution function.
{"title":"Bivariate Weighted Residual and Past Entropies","authors":"G. Rajesh, E. I. Abdul-Sathar, R. Nair","doi":"10.14490/JJSS.46.165","DOIUrl":"https://doi.org/10.14490/JJSS.46.165","url":null,"abstract":"The weighted entropy introduced by Belis and Guiasu (1968) is viewed as a measure of uncertainty. Di Crescenzo and Longobardi (2006) proposed dynamic form of these measure namely weighted residual (WRE) and past entropies (WPE). In this paper, we extend the definition of weighted residual and past entropies to bivariate setup and obtain some of its properties. Several properties, including monotonicity and bounds of BWRE and BWRP are obtained. We also look into the problem of extending WRE and WPE for conditionally specified models. Several properties, including bounds of CWRE and CWPE are obtained for conditional distributions. It is shown that the proposed measure uniquely determines the distribution function.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132644086","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}
Based on the Dirichlet process as a prior, we give the Bayes estimate of the estimable parameter of an arbitrary degree, whose form is different from Yamato (1977b). From it, we derive the simple form of the limit of Bayes estimate as the parameter of the Dirichlet process tends to zero.
{"title":"Limiting Bayes Estimates of Estimable Parameters Based on Dirichlet Processes","authors":"Hajime Yamato","doi":"10.14490/JJSS.46.155","DOIUrl":"https://doi.org/10.14490/JJSS.46.155","url":null,"abstract":"Based on the Dirichlet process as a prior, we give the Bayes estimate of the estimable parameter of an arbitrary degree, whose form is different from Yamato (1977b). From it, we derive the simple form of the limit of Bayes estimate as the parameter of the Dirichlet process tends to zero.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"212 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114050006","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}
For the analysis of square contingency tables with ordered categories, it may be useful for applying some kinds of asymmetry model when the symmetry model does not hold. Tahata and Tomizawa (2011) considered the linear asymmetry model. In the present paper, the extended linear asymmetry model is proposed. The model indicates that the log-odds of symmetric cells are expressed as polynomial function of parameter. Also, the symmetry model is separated into two models and the relationship between test statistics is given.
{"title":"Extended Linear Asymmetry Model and Separation of Symmetry for Square Contingency Tables","authors":"Kouji Tahata, Masato Naganawa, S. Tomizawa","doi":"10.14490/JJSS.46.189","DOIUrl":"https://doi.org/10.14490/JJSS.46.189","url":null,"abstract":"For the analysis of square contingency tables with ordered categories, it may be useful for applying some kinds of asymmetry model when the symmetry model does not hold. Tahata and Tomizawa (2011) considered the linear asymmetry model. In the present paper, the extended linear asymmetry model is proposed. The model indicates that the log-odds of symmetric cells are expressed as polynomial function of parameter. Also, the symmetry model is separated into two models and the relationship between test statistics is given.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131205931","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}
For a one-sided truncated exponential family of distributions with a truncation parameter and a natural parameter as a nuisance parameter, it is shown by Akahira and Ohyauchi (2016) that the second order asymptotic loss of a bias-adjusted maximum likelihood estimator (MLE) of a truncation parameter for unknown natural parameter relative to a bias-adjusted MLE of a truncation parameter for known natural parameter is obtained. In this paper, in a similar way to Akahira and Ohyauchi (2016), for a two-sided truncated exponential family of distributions with a natural parameter and lower and upper truncation parameters, the stochastic expansions of the bias-adjusted MLE of an upper truncation parameter for known natural and lower truncation parameters, the bias-adjusted MLE of an upper truncation parameter for unknown natural parameter and known lower truncation parameter and the bias-adjusted MLE of an upper truncation parameter for unknown natural and lower truncation parameters are derived, their asymptotic variances are given, and the second order asymptotic losses of the MLEs of an upper truncation parameter for unknown natural parameter and known/unknown lower truncation parameter relative to the MLE of an upper truncation parameter for known natural and lower truncation parameters are also obtained. Further, some examples including an upper-truncated Pareto case are given.
{"title":"Second Order Asymptotic Loss of the MLE of a Truncation Parameter for a Two-Sided Truncated Exponential Family of Distributions","authors":"M. Akahira, N. Ohyauchi","doi":"10.14490/JJSS.46.27","DOIUrl":"https://doi.org/10.14490/JJSS.46.27","url":null,"abstract":"For a one-sided truncated exponential family of distributions with a truncation parameter and a natural parameter as a nuisance parameter, it is shown by Akahira and Ohyauchi (2016) that the second order asymptotic loss of a bias-adjusted maximum likelihood estimator (MLE) of a truncation parameter for unknown natural parameter relative to a bias-adjusted MLE of a truncation parameter for known natural parameter is obtained. In this paper, in a similar way to Akahira and Ohyauchi (2016), for a two-sided truncated exponential family of distributions with a natural parameter and lower and upper truncation parameters, the stochastic expansions of the bias-adjusted MLE of an upper truncation parameter for known natural and lower truncation parameters, the bias-adjusted MLE of an upper truncation parameter for unknown natural parameter and known lower truncation parameter and the bias-adjusted MLE of an upper truncation parameter for unknown natural and lower truncation parameters are derived, their asymptotic variances are given, and the second order asymptotic losses of the MLEs of an upper truncation parameter for unknown natural parameter and known/unknown lower truncation parameter relative to the MLE of an upper truncation parameter for known natural and lower truncation parameters are also obtained. Further, some examples including an upper-truncated Pareto case are given.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114071935","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}
For a one-sided truncated exponential family of distributions with a truncation parameter γ and a natural parameter θ as a nuisance parameter, the stochastic expansions of the Bayes estimator ˆ γ B,θ when θ is known and the Bayes estimator ˆ γ B, ˆ θ ML plugging the maximum likelihood estimator (MLE) ˆ θ ML in θ of ˆ γ B,θ when θ is unknown are derived. The second order asymptotic loss of ˆ γ B, ˆ θ ML relative to ˆ γ B,θ is also obtained through their asymptotic variances. Further, it is shown that ˆ γ B,θ and ˆ γ B, ˆ θ ML are second order asymptotically equivalent to the bias-adjusted MLEs ˆ γ ML ∗ ,θ and ˆ γ ML ∗ when θ is known and when θ is unknown, respectively. Some examples are also given.
{"title":"Second Order Asymptotic Variance of the Bayes Estimator of a Truncation Parameter for a One-Sided Truncated Exponential Family of Distributions","authors":"M. Akahira","doi":"10.14490/JJSS.46.81","DOIUrl":"https://doi.org/10.14490/JJSS.46.81","url":null,"abstract":"For a one-sided truncated exponential family of distributions with a truncation parameter γ and a natural parameter θ as a nuisance parameter, the stochastic expansions of the Bayes estimator ˆ γ B,θ when θ is known and the Bayes estimator ˆ γ B, ˆ θ ML plugging the maximum likelihood estimator (MLE) ˆ θ ML in θ of ˆ γ B,θ when θ is unknown are derived. The second order asymptotic loss of ˆ γ B, ˆ θ ML relative to ˆ γ B,θ is also obtained through their asymptotic variances. Further, it is shown that ˆ γ B,θ and ˆ γ B, ˆ θ ML are second order asymptotically equivalent to the bias-adjusted MLEs ˆ γ ML ∗ ,θ and ˆ γ ML ∗ when θ is known and when θ is unknown, respectively. Some examples are also given.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130472709","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 is concerned with the null distribution of the likelihood ratio test statistic −2 log Λ for testing the adequacy of a random-effects covariance structure in a parallel profile model. It is known that the null distribution of −2 log Λ converges to χf or 0.5χf + 0.5χf+1 when the sample size tends to infinity. In order to extend this result, we derive asymptotic expansions of the null distribution of −2 log Λ. The accuracy of the approximations based on the limiting distribution and an asymptotic expansion are compared through numerical experiments.
{"title":"Asymptotic Expansions of the Null Distribution of the LR Test Statistic for Random-Effects Covariance Structure in a Parallel Profile Model","authors":"Yu Inatsu, H. Wakaki","doi":"10.14490/JJSS.46.51","DOIUrl":"https://doi.org/10.14490/JJSS.46.51","url":null,"abstract":"This paper is concerned with the null distribution of the likelihood ratio test statistic −2 log Λ for testing the adequacy of a random-effects covariance structure in a parallel profile model. It is known that the null distribution of −2 log Λ converges to χf or 0.5χf + 0.5χf+1 when the sample size tends to infinity. In order to extend this result, we derive asymptotic expansions of the null distribution of −2 log Λ. The accuracy of the approximations based on the limiting distribution and an asymptotic expansion are compared through numerical experiments.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128423258","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}
G. Rajesh, E. I. Abdul-Sathar, K. V. Reshmi, K. Nair
The Cumulative Residual Entropy (CRE), introduced by Rao et al. (2004), is viewed as a dynamic measure of uncertainty. Recently Asadi and Zohrevand (2007) proposed a dynamic form for the CRE, namely Dynamic Cumulative Residual Entropy (DCRE), and has discussed some of its properties. In this paper, we look into the problem of extending this concept to the conditionally specified models and study various properties of the new measures. We also propose nonparametric estimation for the new measures defined and performance of the estimators are compared using a simulation study.
{"title":"The Conditional Dynamic Cumulative Residual Entropy","authors":"G. Rajesh, E. I. Abdul-Sathar, K. V. Reshmi, K. Nair","doi":"10.14490/JJSS.45.99","DOIUrl":"https://doi.org/10.14490/JJSS.45.99","url":null,"abstract":"The Cumulative Residual Entropy (CRE), introduced by Rao et al. (2004), is viewed as a dynamic measure of uncertainty. Recently Asadi and Zohrevand (2007) proposed a dynamic form for the CRE, namely Dynamic Cumulative Residual Entropy (DCRE), and has discussed some of its properties. In this paper, we look into the problem of extending this concept to the conditionally specified models and study various properties of the new measures. We also propose nonparametric estimation for the new measures defined and performance of the estimators are compared using a simulation study.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124151050","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 Hodrick-Prescott (HP) filter is a popular econometric tool for estimating the trend component of a given time series. Paige and Trindade (2010) present a ridge regression representation of the HP filter, which enhances our understanding of the filter. Schlicht (2005) presents another ridge regression representation of the HP filter. In this paper, we aim to generalize their results. In addition, we present an orthogonal decomposition of the generalized HP and newly introduce the pure generalized HP filter.
{"title":"Ridge Regression Representations of the Generalized Hodrick-Prescott Filter","authors":"Hiroshi Yamada","doi":"10.14490/JJSS.45.121","DOIUrl":"https://doi.org/10.14490/JJSS.45.121","url":null,"abstract":"The Hodrick-Prescott (HP) filter is a popular econometric tool for estimating the trend component of a given time series. Paige and Trindade (2010) present a ridge regression representation of the HP filter, which enhances our understanding of the filter. Schlicht (2005) presents another ridge regression representation of the HP filter. In this paper, we aim to generalize their results. In addition, we present an orthogonal decomposition of the generalized HP and newly introduce the pure generalized HP filter.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128219753","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 method for estimating the parameters of contingency table models, which is motivated by a geometrical idea. Our method—bisector regression for contingency tables (BRCT)—is based on a nested structure of contingency table models. Our method estimates parameters corresponding to the interactions of lower orders after estimating or eliminating those of higher orders. BRCTgenerates a sequence of parameter estimates, each element of which represents a model and a parameter estimate. The length of the sequence is equal to the number of parameters, which is much smaller than the total number of models. We describe the BRCTalgorithm and show an example. We provide explanations for two cases: (a) two factors and (b) K factors.
{"title":"An Estimation Procedure for Contingency Table Models Based on Nested Geometry","authors":"Yoshihiro Hirose, F. Komaki","doi":"10.14490/JJSS.45.57","DOIUrl":"https://doi.org/10.14490/JJSS.45.57","url":null,"abstract":"We propose a method for estimating the parameters of contingency table models, which is motivated by a geometrical idea. Our method—bisector regression for contingency tables (BRCT)—is based on a nested structure of contingency table models. Our method estimates parameters corresponding to the interactions of lower orders after estimating or eliminating those of higher orders. BRCTgenerates a sequence of parameter estimates, each element of which represents a model and a parameter estimate. The length of the sequence is equal to the number of parameters, which is much smaller than the total number of models. We describe the BRCTalgorithm and show an example. We provide explanations for two cases: (a) two factors and (b) K factors.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115251516","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 clarify conditions for consistency of a log-likelihood-based information criterion in multivariate linear regression models with a normality assumption. Although a normality is assumed to the distribution of the candidate model, we frame the situation as that the assumption of normality may be violated. The conditions for consistency are derived from two types of asymptotic theory; one is based on a large-sample asymptotic framework in which only the sample size approaches∞, and the other is based on a high-dimensional asymptotic framework in which the sample size and the dimension of the vector of response variables simultaneously approach ∞. In both cases, our results are free of the influence of nonnormality in the true distribution.
{"title":"Conditions for Consistency of a Log-Likelihood-Based Information Criterion in Normal Multivariate Linear Regression Models under the Violation of the Normality Assumption","authors":"H. Yanagihara","doi":"10.14490/JJSS.45.21","DOIUrl":"https://doi.org/10.14490/JJSS.45.21","url":null,"abstract":"In this paper, we clarify conditions for consistency of a log-likelihood-based information criterion in multivariate linear regression models with a normality assumption. Although a normality is assumed to the distribution of the candidate model, we frame the situation as that the assumption of normality may be violated. The conditions for consistency are derived from two types of asymptotic theory; one is based on a large-sample asymptotic framework in which only the sample size approaches∞, and the other is based on a high-dimensional asymptotic framework in which the sample size and the dimension of the vector of response variables simultaneously approach ∞. In both cases, our results are free of the influence of nonnormality in the true distribution.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131914829","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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