In this paper, we consider the general linear hypothesis testing (GLHT) problem in heteroscedastic one-way MANOVA. The well-known Wald-type test statistic is used. Its null distribution is approximated by a Hotelling ...In this paper, we consider the general linear hypothesis testing (GLHT) problem in heteroscedastic one-way MANOVA. The well-known Wald-type test statistic is used. Its null distribution is approximated by a Hotelling T2 distribution with one parameter estimated from the data, resulting in the so-called approximate Hotelling T2 (AHT) test. The AHT test is shown to be invariant under affine transformation, different choices of the contrast matrix specifying the same hypothesis, and different labeling schemes of the mean vectors. The AHT test can be simply conducted using the usual F-distribution. Simulation studies and real data applications show that the AHT test substantially outperforms the test of [1] and is comparable to the parametric bootstrap (PB) test of [2] for the multivariate k-sample Behrens-Fisher problem which is a special case of the GLHT problem in heteroscedastic one-way MANOVA.展开更多
The TEEOF method that expands temporally is used to conduct a diagnostic study of the variation patterns of 1, 3, 6 and 10 years with regard to mean air temperature over the globe and Southern and Northern Hemispheres...The TEEOF method that expands temporally is used to conduct a diagnostic study of the variation patterns of 1, 3, 6 and 10 years with regard to mean air temperature over the globe and Southern and Northern Hemispheres over the course of 100 years. The results show that the first mode of TEEOF takes up more than 50% in the total variance, with each of the first mode in the interannual oscillations generally standing for annually varying patterns which are related with climate and reflecting long-term tendency of change in air temperature. It is particularly true for the first mode on the 10-year scale, which shows an obvious ascending trend concerning the temperature in winter and consistently the primary component of time goes in a way that is very close to the sequence of actual temperature. Apart from the first mode of all time sections of TEEOF for the globe and the two hemispheres and the second mode of the 1-year TEEOF, interannual variation described by other characteristic vectors are showing various patterns, with corresponding primary components having relation with long-term variability of specific interannual quasi-periodic oscillation structures. A 2T test applied to the annual variation pattern shows that the abrupt changes for the Southern Hemisphere and the globe come closer to the result of a uni-element t test for mean temperature than those for the Northern Hemisphere do. It indicates that the 2Ttest, when carried out with patterns of multiple variables, seems more reasonable than the t test with single elements.展开更多
Detecting differential expression of genes in genom research(e.g.,2019-nCoV)is not uncommon,due to the cost only small sample is employed to estimate a large number of variances(or their inverse)of variables simultane...Detecting differential expression of genes in genom research(e.g.,2019-nCoV)is not uncommon,due to the cost only small sample is employed to estimate a large number of variances(or their inverse)of variables simultaneously.However,the commonly used approaches perform unreliable.Borrowing information across different variables or priori information of variables,shrinkage estimation approaches are proposed and some optimal shrinkage estimators are obtained in the sense of asymptotic.In this paper,we focus on the setting of small sample and a likelihood-unbiased estimator for power of variances is given under the assumption that the variances are chi-squared distribution.Simulation reports show that the likelihood-unbiased estimators for variances and their inverse perform very well.In addition,application comparison and real data analysis indicate that the proposed estimator also works well.展开更多
For several decades, much attention has been paid to the two-sample Behrens-Fisher (BF) problem which tests the equality of the means or mean vectors of two normal populations with unequal variance/covariance structur...For several decades, much attention has been paid to the two-sample Behrens-Fisher (BF) problem which tests the equality of the means or mean vectors of two normal populations with unequal variance/covariance structures. Little work, however, has been done for the k-sample BF problem for high dimensional data which tests the equality of the mean vectors of several high-dimensional normal populations with unequal covariance structures. In this paper we study this challenging problem via extending the famous Scheffe’s transformation method, which reduces the k-sample BF problem to a one-sample problem. The induced one-sample problem can be easily tested by the classical Hotelling’s T 2 test when the size of the resulting sample is very large relative to its dimensionality. For high dimensional data, however, the dimensionality of the resulting sample is often very large, and even much larger than its sample size, which makes the classical Hotelling’s T 2 test not powerful or not even well defined. To overcome this difficulty, we propose and study an L 2-norm based test. The asymptotic powers of the proposed L 2-norm based test and Hotelling’s T 2 test are derived and theoretically compared. Methods for implementing the L 2-norm based test are described. Simulation studies are conducted to compare the L 2-norm based test and Hotelling’s T 2 test when the latter can be well defined, and to compare the proposed implementation methods for the L 2-norm based test otherwise. The methodologies are motivated and illustrated by a real data example.展开更多
文摘In this paper, we consider the general linear hypothesis testing (GLHT) problem in heteroscedastic one-way MANOVA. The well-known Wald-type test statistic is used. Its null distribution is approximated by a Hotelling T2 distribution with one parameter estimated from the data, resulting in the so-called approximate Hotelling T2 (AHT) test. The AHT test is shown to be invariant under affine transformation, different choices of the contrast matrix specifying the same hypothesis, and different labeling schemes of the mean vectors. The AHT test can be simply conducted using the usual F-distribution. Simulation studies and real data applications show that the AHT test substantially outperforms the test of [1] and is comparable to the parametric bootstrap (PB) test of [2] for the multivariate k-sample Behrens-Fisher problem which is a special case of the GLHT problem in heteroscedastic one-way MANOVA.
文摘The TEEOF method that expands temporally is used to conduct a diagnostic study of the variation patterns of 1, 3, 6 and 10 years with regard to mean air temperature over the globe and Southern and Northern Hemispheres over the course of 100 years. The results show that the first mode of TEEOF takes up more than 50% in the total variance, with each of the first mode in the interannual oscillations generally standing for annually varying patterns which are related with climate and reflecting long-term tendency of change in air temperature. It is particularly true for the first mode on the 10-year scale, which shows an obvious ascending trend concerning the temperature in winter and consistently the primary component of time goes in a way that is very close to the sequence of actual temperature. Apart from the first mode of all time sections of TEEOF for the globe and the two hemispheres and the second mode of the 1-year TEEOF, interannual variation described by other characteristic vectors are showing various patterns, with corresponding primary components having relation with long-term variability of specific interannual quasi-periodic oscillation structures. A 2T test applied to the annual variation pattern shows that the abrupt changes for the Southern Hemisphere and the globe come closer to the result of a uni-element t test for mean temperature than those for the Northern Hemisphere do. It indicates that the 2Ttest, when carried out with patterns of multiple variables, seems more reasonable than the t test with single elements.
基金Supported by the National Natural Science Foundation of China(11971433)First Class Discipline of Zhejiang-A(Zhejiang Gongshang University-Statistics)Hunan Soft Science Research Project(2012ZK3064)
文摘Detecting differential expression of genes in genom research(e.g.,2019-nCoV)is not uncommon,due to the cost only small sample is employed to estimate a large number of variances(or their inverse)of variables simultaneously.However,the commonly used approaches perform unreliable.Borrowing information across different variables or priori information of variables,shrinkage estimation approaches are proposed and some optimal shrinkage estimators are obtained in the sense of asymptotic.In this paper,we focus on the setting of small sample and a likelihood-unbiased estimator for power of variances is given under the assumption that the variances are chi-squared distribution.Simulation reports show that the likelihood-unbiased estimators for variances and their inverse perform very well.In addition,application comparison and real data analysis indicate that the proposed estimator also works well.
基金supported by the National University of Singapore Academic Research Grant (Grant No. R-155-000-085-112)
文摘For several decades, much attention has been paid to the two-sample Behrens-Fisher (BF) problem which tests the equality of the means or mean vectors of two normal populations with unequal variance/covariance structures. Little work, however, has been done for the k-sample BF problem for high dimensional data which tests the equality of the mean vectors of several high-dimensional normal populations with unequal covariance structures. In this paper we study this challenging problem via extending the famous Scheffe’s transformation method, which reduces the k-sample BF problem to a one-sample problem. The induced one-sample problem can be easily tested by the classical Hotelling’s T 2 test when the size of the resulting sample is very large relative to its dimensionality. For high dimensional data, however, the dimensionality of the resulting sample is often very large, and even much larger than its sample size, which makes the classical Hotelling’s T 2 test not powerful or not even well defined. To overcome this difficulty, we propose and study an L 2-norm based test. The asymptotic powers of the proposed L 2-norm based test and Hotelling’s T 2 test are derived and theoretically compared. Methods for implementing the L 2-norm based test are described. Simulation studies are conducted to compare the L 2-norm based test and Hotelling’s T 2 test when the latter can be well defined, and to compare the proposed implementation methods for the L 2-norm based test otherwise. The methodologies are motivated and illustrated by a real data example.
