In this paper, we construct a confidence region for the efficient frontier assuming the asset returns to be matrix elliptically contoured distributed. Our results extend the findings of Bodnar and Schmid (2009) to the non-normal distributed asset returns. In order to correct the overoptimism of the sample efficient frontier documented in Siegel and Woodgate (2007), the unbiased estimator of the efficient frontier is suggested. Moreover, we derive an exact overall F -test for the efficient frontier in elliptical models.
{"title":"CONSTRUCTION AND INFERENCES OF THE EFFICIENT FRONTIER IN ELLIPTICAL MODELS","authors":"Taras Bodnar, Arjun K. Gupta","doi":"10.14490/JJSS.39.193","DOIUrl":"https://doi.org/10.14490/JJSS.39.193","url":null,"abstract":"In this paper, we construct a confidence region for the efficient frontier assuming the asset returns to be matrix elliptically contoured distributed. Our results extend the findings of Bodnar and Schmid (2009) to the non-normal distributed asset returns. In order to correct the overoptimism of the sample efficient frontier documented in Siegel and Woodgate (2007), the unbiased estimator of the efficient frontier is suggested. Moreover, we derive an exact overall F -test for the efficient frontier in elliptical models.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122467424","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}
AB ayesian method for diagnosing a hypothesis is proposed in terms of the optimum Bayesian predictor under the e-divergence loss. We introduce a predictive credible region as a modified version of a posterior credible region. The predictive credible region is closely related to the complement of the rejection region of the likelihood ratio test in the frequentist context. As an application we revisit the controversy regarding Lindley’s paradox, and observe satisfactory performance of the proposed credible region in contrast to the Bayes factor. Another important application concerns a method for analyzing additional evidence when a hypothesis is once rejected.
{"title":"PREDICTIVE CREDIBLE REGION FOR BAYESIAN DIAGNOSIS OF A HYPOTHESIS","authors":"T. Yanagimoto, Toshio Ohnishi","doi":"10.14490/JJSS.39.111","DOIUrl":"https://doi.org/10.14490/JJSS.39.111","url":null,"abstract":"AB ayesian method for diagnosing a hypothesis is proposed in terms of the optimum Bayesian predictor under the e-divergence loss. We introduce a predictive credible region as a modified version of a posterior credible region. The predictive credible region is closely related to the complement of the rejection region of the likelihood ratio test in the frequentist context. As an application we revisit the controversy regarding Lindley’s paradox, and observe satisfactory performance of the proposed credible region in contrast to the Bayes factor. Another important application concerns a method for analyzing additional evidence when a hypothesis is once rejected.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129500292","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 unusual rate of convergence diag{√n log(1/hn), √ n, √ nh 1−1/α n }, but the Fisher information matrix is constantly singular as soon as both α and σ are unknown. This implies that the standard asymptotic behavior of the maximum likelihood estimator breaks down, and also that it is in no way obvious whether or not existing results concerning estimators of the stable law in the usual case where hn ≡ h > 0 can maintain the same asymptotic behaviors. In this note we will provide easily computable full-joint estimators of the parameters, which possess asymptotic normality with a finite and nondegenerate asymptotic covariance matrix, thereby enabling us to construct a joint confidence region of the three parameters: the rate of convergence of our estimators of θ is diag( √ n, √ n, √ nh 1−1/α n ). Especially, we clarify that a suitable sample-median type statistic γn serves as a rate-efficient estimator of the location γ, and that our procedure of estimating the remaining two parameters is not asymptotically influenced by plugging in γn, even if the convergence rate of γn is slower than the other two (namely, even if α ∈ (1, 2)). Finite-sample behaviors of our estimators are investigated through several simulation experiments.
{"title":"Joint Estimation of Discretely Observed Stable Lévy Processes with Symmetric Lévy Density","authors":"Hiroki Masuda","doi":"10.14490/JJSS.39.49","DOIUrl":"https://doi.org/10.14490/JJSS.39.49","url":null,"abstract":"the unusual rate of convergence diag{√n log(1/hn), √ n, √ nh 1−1/α n }, but the Fisher information matrix is constantly singular as soon as both α and σ are unknown. This implies that the standard asymptotic behavior of the maximum likelihood estimator breaks down, and also that it is in no way obvious whether or not existing results concerning estimators of the stable law in the usual case where hn ≡ h > 0 can maintain the same asymptotic behaviors. In this note we will provide easily computable full-joint estimators of the parameters, which possess asymptotic normality with a finite and nondegenerate asymptotic covariance matrix, thereby enabling us to construct a joint confidence region of the three parameters: the rate of convergence of our estimators of θ is diag( √ n, √ n, √ nh 1−1/α n ). Especially, we clarify that a suitable sample-median type statistic γn serves as a rate-efficient estimator of the location γ, and that our procedure of estimating the remaining two parameters is not asymptotically influenced by plugging in γn, even if the convergence rate of γn is slower than the other two (namely, even if α ∈ (1, 2)). Finite-sample behaviors of our estimators are investigated through several simulation experiments.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121060325","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 analyze the dynamic labor demand structure of large Japanese firms. We propose a new dynamic model which explicitly considers the asymmetric behavior of the firms between decreasing and increasing regimes. The model modifies the ordinary partial adjustment and switching cost models. The model is a Tobit-type model; that is, the employment strategies and desired levels of labor are determined by latent variables. We estimate the model using the data augmentation algorithm, which is a Bayesian simulation method. We apply the model to the panel data constructed from financial reports of large Japanese manufacturing firms. When asymmetric adjustment costs are included in the model, we find that: i) the hiring cost does not become lower even if lay-offs and dismissals are easier, and ii) employment strategies differ among the industrial sectors even if their cost structures are similar.
{"title":"Bayesian Estimation of an Asymmetric Employment Adjustment Model","authors":"A. Matsumoto, Hisayuki Hara, K. Nawata","doi":"10.14490/JJSS.39.29","DOIUrl":"https://doi.org/10.14490/JJSS.39.29","url":null,"abstract":"In this paper, we analyze the dynamic labor demand structure of large Japanese firms. We propose a new dynamic model which explicitly considers the asymmetric behavior of the firms between decreasing and increasing regimes. The model modifies the ordinary partial adjustment and switching cost models. The model is a Tobit-type model; that is, the employment strategies and desired levels of labor are determined by latent variables. We estimate the model using the data augmentation algorithm, which is a Bayesian simulation method. We apply the model to the panel data constructed from financial reports of large Japanese manufacturing firms. When asymmetric adjustment costs are included in the model, we find that: i) the hiring cost does not become lower even if lay-offs and dismissals are easier, and ii) employment strategies differ among the industrial sectors even if their cost structures are similar.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125200479","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 the estimation problem of the population density of organisms in an area. We formulate this biometric problem as a variant of the sequential estimation problem of the intensity of the Poisson process and propose a method for simultaneously optimizing both the decision of an observation subarea and the estimation. By request from the application side, we define the equivariance of a procedure composed of a decision rule of an observation subarea and an estimator of population density under the scale transformations of an observation area, and then construct a scale-equivariant procedure. As a result of using the invariance principle, we utilize the framework of statistical decision theory, not the conventional framework of sequential estimation using asymptotic methods, and discuss the admissibility and minimaxity of our proposed procedure.
{"title":"OBSERVATION SUBAREA DECISION AND POPULATION DENSITY ESTIMATION BY SPACE SCALE-INVARIANCE","authors":"H. Kishino","doi":"10.14490/JJSS.39.77","DOIUrl":"https://doi.org/10.14490/JJSS.39.77","url":null,"abstract":"In this paper, we consider the estimation problem of the population density of organisms in an area. We formulate this biometric problem as a variant of the sequential estimation problem of the intensity of the Poisson process and propose a method for simultaneously optimizing both the decision of an observation subarea and the estimation. By request from the application side, we define the equivariance of a procedure composed of a decision rule of an observation subarea and an estimator of population density under the scale transformations of an observation area, and then construct a scale-equivariant procedure. As a result of using the invariance principle, we utilize the framework of statistical decision theory, not the conventional framework of sequential estimation using asymptotic methods, and discuss the admissibility and minimaxity of our proposed procedure.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125249154","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 problem of discriminating between two competing simple linear regression models MI and MII is discussed in this paper. The model MI is nested within the model MII with a common linear term being present in both models with respect to an explanatory variable and an additional quadratic term with respect to the same explanatory variable being present in MII. The first criterion function for discrimination between MI and MII is in terms of minimizing the variance of the least squares estimated coefficient of the quadratic term. A lower bound is obtained for this variance. Two designs are presented satisfying this lower bound in two experimental regions. New criterion functions of discriminating between MI and MII are given based on maximizing the difference between the fitted values of n observations under MI and MII and maximizing the difference between the predicted values under MI and MII. Several results are obtained for demonstrating the performances of these designs under two new criterion functions. We also present five general classes of designs and demonstrate their sharp relative performances with respect to our criterion functions. Some results for discriminating between two competing general linear models MIII and MIV are also given.
{"title":"Design and Inference for Discriminating Between Two Competing Linear Regression Models","authors":"Subir Ghosh, J. López–Fidalgo, Rupam Pal","doi":"10.14490/JJSS.39.15","DOIUrl":"https://doi.org/10.14490/JJSS.39.15","url":null,"abstract":"The problem of discriminating between two competing simple linear regression models MI and MII is discussed in this paper. The model MI is nested within the model MII with a common linear term being present in both models with respect to an explanatory variable and an additional quadratic term with respect to the same explanatory variable being present in MII. The first criterion function for discrimination between MI and MII is in terms of minimizing the variance of the least squares estimated coefficient of the quadratic term. A lower bound is obtained for this variance. Two designs are presented satisfying this lower bound in two experimental regions. New criterion functions of discriminating between MI and MII are given based on maximizing the difference between the fitted values of n observations under MI and MII and maximizing the difference between the predicted values under MI and MII. Several results are obtained for demonstrating the performances of these designs under two new criterion functions. We also present five general classes of designs and demonstrate their sharp relative performances with respect to our criterion functions. Some results for discriminating between two competing general linear models MIII and MIV are also given.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131018222","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 describe an algorithm for extracting as much information as possible from pooling experiments for library screening based on the concave-convex procedure (CCCP). Called the CCCP pool result decoder (CCPD), it is a positive clone de- tecting algorithm. Its performance is compared, by simulation, with the Bayesian network pool result decoder (BNPD) proposed by Uehara and Jimbo and the Markov chain pool result decoder (MCPD) proposed by Knill et al. in 1996. To find a few positives among a large number of items, one can use group testing. In group testing, multiple items are assayed in groups. If a group has a negative outcome, all items contained in it are negative. This can reduce the total number of tests. On the other hand, if a group is positive, we know that the group contains at least one positive item. By designing many kinds of groups and by testing each of them, we obtain the results for all groups. After knowing the group results, we may be able to estimate which items are likely to be positive. For each of such items, we apply individual tests to determine whether it is positive or negative. Group testing has been used in medical, chemical, and electrical testing; drug screening; pollution control; multiaccess channel communication; and recently in gene assays like clone library screening, protein-protein interaction tests, and other subjects. See for example Du and Hwang (1999), Schliep and Rahmann (2006), Thierry-Mieg (2006), Klau et al. (2007), Thierry-Mieg and Bailly (2008). In this paper we restrict ourselves to group testing for DNA library screening to give a concrete image of testing and to consider a specialized problem in clone library screening. However, the algorithm given in this paper can be applied to any other fields of group testing. In DNA library screening experiments there are a large number of clones, which are short strings of nucleotides A, T, G and C. Through the use of high-quality gene libraries, the study of gene functions has been developed into a very important research field. The gene libraries are obtained from extensive testing and screening of DNA clones. For each clone,
我们描述了一种基于凹凸过程(CCCP)的从池化实验中提取尽可能多的信息用于图书馆筛选的算法。CCCP池结果解码器(CCPD)是一种正克隆检测算法。通过仿真，将其性能与Uehara和Jimbo提出的贝叶斯网络池结果解码器(BNPD)和Knill等人1996年提出的马尔可夫链池结果解码器(MCPD)进行了比较。为了在大量的项目中找到一些阳性的，可以使用群体测试。在分组测试中，多个项目被分组分析。如果一个组的结果是否定的，那么其中包含的所有项都是否定的。这可以减少测试的总数。另一方面，如果一个组是正的，我们知道这个组至少包含一个正的元素。通过设计多种组，并对每个组进行测试，得到了所有组的结果。在知道分组结果之后，我们可能能够估计哪些项目可能是积极的。对于每一个这样的项目，我们应用单独的测试来确定它是阳性还是阴性。在医疗、化学和电气测试中使用了分组测试;药物筛选;污染控制;多址信道通信;最近在基因分析，如克隆文库筛选，蛋白质-蛋白质相互作用测试，和其他科目。参见Du and Hwang (1999)， Schliep and Rahmann (2006)， Thierry-Mieg (2006)， Klau et al. (2007)， Thierry-Mieg and Bailly(2008)。本文将DNA文库筛选限定为群体检测，以给出具体的检测形象，并考虑克隆文库筛选中的一个专门问题。然而，本文给出的算法可以应用于群测试的任何其他领域。在DNA文库筛选实验中存在大量的克隆，这些克隆是核苷酸a、T、G和c的短链，通过使用高质量的基因库，基因功能的研究已经发展成为一个非常重要的研究领域。基因库是从DNA克隆的广泛测试和筛选中获得的。对于每一个克隆体，
{"title":"A POSITIVE DETECTING ALGORITHM FOR DNA LIBRARY SCREENING BASED ON CCCP","authors":"Hiroaki Uehara, Masakazu Jimbo","doi":"10.14490/JJSS.39.89","DOIUrl":"https://doi.org/10.14490/JJSS.39.89","url":null,"abstract":"We describe an algorithm for extracting as much information as possible from pooling experiments for library screening based on the concave-convex procedure (CCCP). Called the CCCP pool result decoder (CCPD), it is a positive clone de- tecting algorithm. Its performance is compared, by simulation, with the Bayesian network pool result decoder (BNPD) proposed by Uehara and Jimbo and the Markov chain pool result decoder (MCPD) proposed by Knill et al. in 1996. To find a few positives among a large number of items, one can use group testing. In group testing, multiple items are assayed in groups. If a group has a negative outcome, all items contained in it are negative. This can reduce the total number of tests. On the other hand, if a group is positive, we know that the group contains at least one positive item. By designing many kinds of groups and by testing each of them, we obtain the results for all groups. After knowing the group results, we may be able to estimate which items are likely to be positive. For each of such items, we apply individual tests to determine whether it is positive or negative. Group testing has been used in medical, chemical, and electrical testing; drug screening; pollution control; multiaccess channel communication; and recently in gene assays like clone library screening, protein-protein interaction tests, and other subjects. See for example Du and Hwang (1999), Schliep and Rahmann (2006), Thierry-Mieg (2006), Klau et al. (2007), Thierry-Mieg and Bailly (2008). In this paper we restrict ourselves to group testing for DNA library screening to give a concrete image of testing and to consider a specialized problem in clone library screening. However, the algorithm given in this paper can be applied to any other fields of group testing. In DNA library screening experiments there are a large number of clones, which are short strings of nucleotides A, T, G and C. Through the use of high-quality gene libraries, the study of gene functions has been developed into a very important research field. The gene libraries are obtained from extensive testing and screening of DNA clones. For each clone,","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129887731","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, the efficiencies of various second order response surface designs, like second order rotatable designs (SORD), second order slope rotatable designs (SOSRD), SORDs with an equi-spaced doses design, and SOSRDs with an equispaced doses design using symmetrical unequal block arrangements (SUBA) with two unequal block sizes, are studied for the estimation of responses and slopes at different points (central, axial, cube corner points) on second order response surface designs. The study is done because in some cases a SORD, and SOSRD constructed by using a SUBA with two unequal block sizes have fewer design points compared to those constructed by other methods.
{"title":"ON THE EFFICIENCIES OF SECOND ORDER RESPONSE SURFACE DESIGNS FOR THE ESTIMATION OF RESPONSES AND SLOPES USING SYMMETRICAL UNEQUAL BLOCK ARRANGEMENTS WITH TWO UNEQUAL BLOCK SIZES","authors":"B. Victorbabu, V. Vasundharadevi","doi":"10.14490/JJSS.39.1","DOIUrl":"https://doi.org/10.14490/JJSS.39.1","url":null,"abstract":"In this paper, the efficiencies of various second order response surface designs, like second order rotatable designs (SORD), second order slope rotatable designs (SOSRD), SORDs with an equi-spaced doses design, and SOSRDs with an equispaced doses design using symmetrical unequal block arrangements (SUBA) with two unequal block sizes, are studied for the estimation of responses and slopes at different points (central, axial, cube corner points) on second order response surface designs. The study is done because in some cases a SORD, and SOSRD constructed by using a SUBA with two unequal block sizes have fewer design points compared to those constructed by other methods.","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126488919","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}
Inspired by Doob’s suggestion in 1949, Donsker (1952) proved the Kolmogorov-Smirnov theorem in an elegant way via the functional central limit theorem (the invariance principle). In that theorem, the underlying distribution is assumed to be a continuous distribution. On the other hand, real data in practice is always given in a discretized (rounded) form. In this paper, we establish an invariance principle for discretized data in the fashion of the modern empirical process theory to obtain a (right) Kolmogorov-Smirnov test for discretized data. To illustrate our problem let us begin with the most basic example. We denote by F0 the uniform distribution on [0, 1]. Let {X1, . . . , Xn} be an independent sequence of [0, 1]-valued random variables with the common law F0. Set δn = 0.01. Suppose that we can actually observe the data {X i } which is discretized (rounded) up to δn: X1 = 0.67774205 X 1 = 0.68 X2 = 0.81124449 X 2 = 0.81 · · · Xn = 0.61694806 X n = 0.62. We denote by F̂n and F̂ n the empirical distribution functions of {X1, . . . , Xn} and {X 1 , . . . , X n}, respectively. Then, the Kolmogorov-Smirnov statistic Dn = sup t∈[0,1] n|F̂n(t) − F0(t)| converges in distribution to supu∈[0,1] |B◦(u)|, where u ❀ B◦(u) is a standard Brownian bridge. On the other hand, as we will show below, if δn = o(n −1/2), the test statistic D n = sup t∈[0,1] n|F̂ n(t) − F0(t)|
{"title":"DONSKER'S THEOREM FOR DISCRETIZED DATA","authors":"Y. Nishiyama","doi":"10.14490/JJSS.38.505","DOIUrl":"https://doi.org/10.14490/JJSS.38.505","url":null,"abstract":"Inspired by Doob’s suggestion in 1949, Donsker (1952) proved the Kolmogorov-Smirnov theorem in an elegant way via the functional central limit theorem (the invariance principle). In that theorem, the underlying distribution is assumed to be a continuous distribution. On the other hand, real data in practice is always given in a discretized (rounded) form. In this paper, we establish an invariance principle for discretized data in the fashion of the modern empirical process theory to obtain a (right) Kolmogorov-Smirnov test for discretized data. To illustrate our problem let us begin with the most basic example. We denote by F0 the uniform distribution on [0, 1]. Let {X1, . . . , Xn} be an independent sequence of [0, 1]-valued random variables with the common law F0. Set δn = 0.01. Suppose that we can actually observe the data {X i } which is discretized (rounded) up to δn: X1 = 0.67774205 X 1 = 0.68 X2 = 0.81124449 X 2 = 0.81 · · · Xn = 0.61694806 X n = 0.62. We denote by F̂n and F̂ n the empirical distribution functions of {X1, . . . , Xn} and {X 1 , . . . , X n}, respectively. Then, the Kolmogorov-Smirnov statistic Dn = sup t∈[0,1] n|F̂n(t) − F0(t)| converges in distribution to supu∈[0,1] |B◦(u)|, where u ❀ B◦(u) is a standard Brownian bridge. On the other hand, as we will show below, if δn = o(n −1/2), the test statistic D n = sup t∈[0,1] n|F̂ n(t) − F0(t)|","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121154561","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 canonical correlation analysis, canonical vectors are used in the interpretation of the canonical variables. We are interested in the asymptotic representation of the expectation, the variance and the distribution of the canonical vector. In this study, we derive the asymptotic distribution of the canonical vector under nonnormality. To obtain the asymptotic expansion of the canonical vector, we use a perturbation method. In addition, as an example, we show the asymptotic distribution with an elliptical population. In multivariate statistical analysis, the distributions of latent roots and latent vectors of certain symmetric matrices constructed from the sample covariance matrix are important in some cases and have been studied by many authors. These studies can be used as the basis for the canonical correlation analysis, which is an approach that characterizes the correlation structure between two sets of variables. Considering the distribution of the canonical correlation with the assumption of the multivariate normal population, asymptotic expansions of the distributions were studied by Sugiura (1976), Fujikoshi (1977, 1978), Muirhead (1978) and others. The distributions of a function of latent roots of the sample covariance matrix in nonnormal populations were studied by Fujikoshi (1980), Muirhead and Waternaux (1980), Fang and Krishinaiah (1982), Siotani et al. (1985), Seo et al. (1994) and others. The distribution of the canonical vector was studied by Eaton and Tyler (1994), Boik (1998), Anderson (1999), Taskinen et al. (2006) and others. This paper deals with the asymptotic expansion of the canonical vector under nonnormality. Let us denote x =( x � 1, x � 2) � as p + q dimensional variables with mean µ and
在典型相关分析中，典型向量用于解释典型变量。我们感兴趣的是期望的渐近表示，方差和正则向量的分布。本文给出了正则向量在非正态下的渐近分布。为了得到正则向量的渐近展开式，我们使用了摄动方法。此外，作为一个例子，我们给出了椭圆总体的渐近分布。在多元统计分析中，由样本协方差矩阵构造的对称矩阵的隐根和隐向量的分布在某些情况下是很重要的，许多作者已经对此进行了研究。这些研究可以作为典型相关分析的基础，典型相关分析是一种表征两组变量之间相关结构的方法。Sugiura(1976)、Fujikoshi(1977,1978)、Muirhead(1978)等人考虑到典型相关在多元正态总体假设下的分布，研究了典型相关分布的渐近展开。Fujikoshi(1980)、Muirhead和Waternaux(1980)、Fang和Krishinaiah(1982)、Siotani等人(1985)、Seo等人(1994)等研究了样本协方差矩阵潜根函数在非正态总体中的分布。典型向量的分布由Eaton and Tyler(1994)、Boik(1998)、Anderson(1999)、Taskinen et al.(2006)等人研究。本文讨论了正则向量在非正态下的渐近展开式。我们将x =(x′1,x′2)′表示为p + q维变量，其平均值为µand
{"title":"The Efficiency of the Asymptotic Expansion of the Distribution of the Canonical Vector under Nonnormality","authors":"Tomoya Yamada","doi":"10.14490/JJSS.38.451","DOIUrl":"https://doi.org/10.14490/JJSS.38.451","url":null,"abstract":"In canonical correlation analysis, canonical vectors are used in the interpretation of the canonical variables. We are interested in the asymptotic representation of the expectation, the variance and the distribution of the canonical vector. In this study, we derive the asymptotic distribution of the canonical vector under nonnormality. To obtain the asymptotic expansion of the canonical vector, we use a perturbation method. In addition, as an example, we show the asymptotic distribution with an elliptical population. In multivariate statistical analysis, the distributions of latent roots and latent vectors of certain symmetric matrices constructed from the sample covariance matrix are important in some cases and have been studied by many authors. These studies can be used as the basis for the canonical correlation analysis, which is an approach that characterizes the correlation structure between two sets of variables. Considering the distribution of the canonical correlation with the assumption of the multivariate normal population, asymptotic expansions of the distributions were studied by Sugiura (1976), Fujikoshi (1977, 1978), Muirhead (1978) and others. The distributions of a function of latent roots of the sample covariance matrix in nonnormal populations were studied by Fujikoshi (1980), Muirhead and Waternaux (1980), Fang and Krishinaiah (1982), Siotani et al. (1985), Seo et al. (1994) and others. The distribution of the canonical vector was studied by Eaton and Tyler (1994), Boik (1998), Anderson (1999), Taskinen et al. (2006) and others. This paper deals with the asymptotic expansion of the canonical vector under nonnormality. Let us denote x =( x � 1, x � 2) � as p + q dimensional variables with mean µ and","PeriodicalId":326924,"journal":{"name":"Journal of the Japan Statistical Society. Japanese issue","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116727431","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：