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
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.
