查看更多>>摘要：Recently, the shrinkage approach has increased its popularity in theoretical and applied statistics, especially, when point estimators for high-dimensional quantities have to be constructed. A shrinkage estimator is usually obtained by shrinking the sample estimator towards a deterministic target. This allows to reduce the high volatility that is commonly present in the sample estimator by introducing a bias such that the mean-square error of the shrinkage estimator becomes smaller than the one of the corresponding sample estimator. The procedure has shown great advantages especially in the high-dimensional problems where, in general case, the sample estimators are not consistent without imposing structural assumptions on model parameters. In this paper, we review the mostly used shrinkage estimators for the mean vector, covariance and precision matrices. The application in portfolio theory is provided where the weights of optimal portfolios are usually determined as functions of the mean vector and covariance matrix. Furthermore, a test theory on the mean-variance optimality of a given portfolio based on the shrinkage approach is presented as well. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：Many linear dimension reduction methods proposed in the literature can be formulated using an appropriate pair of scatter matrices. The eigen-decomposition of one scatter matrix with respect to another is then often used to determine the dimension of the signal subspace and to separate signal and noise parts of the data. Three popular dimension reduction methods, namely principal component analysis (PCA), fourth order blind identification (FOBI) and sliced inverse regression (SIR) are considered in detail and the first two moments of subsets of the eigenvalues are used to test for the dimension of the signal space. The limiting null distributions of the test statistics are discussed and novel bootstrap strategies are suggested for the small sample cases. In all three cases, consistent test-based estimates of the signal subspace dimension are introduced as well. The asymptotic and bootstrap tests are illustrated in real data examples. (C) 2021 The Author(s). Published by Elsevier Inc.
查看更多>>摘要：A broad review is given of some areas of multivariate analysis that are not frequently emphasized. We start with situations in which underlying distributions are far from multivariate normal form, so that standard methods of multivariate analysis based on covariances are likely to be unsatisfactory. We emphasize the important distinction between internal and external analyses associated with multiple outcomes. A second broad theme relates to multiple outcomes generated by time or spatial series in which long-range dependence operates. Some implications are summarized. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：Principal Components Analysis was developed by Harold Hotelling (1895-1973) in 1933 and Canonical Correlations Analysis in 1936. In this article we trace some of the stages leading up to the development of these procedures, chiefly in the hands of Francis Galton (1822-1911) and Karl Pearson (1857-1936) paying particular attention to the two-variable case developed independently by Julius Ludwig Weisbach (1806-1871) in 1840 and Robert Jackson Adcock (1826-1895) in 1877-78. (c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：Testing high-dimensional means has many applications in scientific research. For instance, it is of great interest to test whether there is a difference of gene expressions between control and treatment groups in genetic studies. This can be formulated as a two-sample mean testing problem. However, the Hotelling T-2 test statistic for the two-sample mean problem is no longer well defined due to singularity of the sample covariance matrix when the sample size is less than the dimension of data. Over the last two decades, the high-dimensional mean testing problem has received considerable attentions in the literature. This paper provides a selective overview of existing testing procedures in the literature. We focus on the motivation of the testing procedures, the insights into how to construct the test statistics and the connections, and comparisons of different methods. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：In this article, the joint best linear unbiased predictors (BLUPs) of two future unobserved order statistics, based on a set of observed order statistics, are developed explicitly. It is shown that these predictors are trace-efficient as well as determinant-efficient BLUPs. More generally, the BLUPs are shown to possess complete mean squared predictive error matrix dominance in the class of all linear unbiased predictors of two future unobserved order statistics. Finally, these results are extended to the case of simultaneous BLUPs of any l future order statistics. Both scale and location-scale family of distributions are considered as the parent distribution for the underlying random variables. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：Scatter matrices generalize the covariance matrix and are useful in many multivariate data analysis methods, including well-known principal component analysis (PCA), which is based on the diagonalization of the covariance matrix. The simultaneous diagonalization of two or more scatter matrices goes beyond PCA and is used more and more often. In this paper, we offer an overview of many methods that are based on a joint diagonalization. These methods range from the unsupervised context with invariant coordinate selection and blind source separation, which includes independent component analysis, to the supervised context with discriminant analysis and sliced inverse regression. They also encompass methods that handle dependent data such as time series or spatial data. (C) 2021 The Author(s). Published by Elsevier Inc.
Kuriki, SatoshiTakemura, AkimichiTaylor, Jonathan E.
23页
查看更多>>摘要：The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then transforms it to the tail probability. In this study, we generalize the tube method to a case in which the variance is not constant. We provide the volume formula for a spherical tube with a non-constant radius in terms of curvature tensors, and the tail probability formula of the maximum of a Gaussian random field with inhomogeneous variance, as well as its Laplace approximation. In particular, the critical radius of the tube is generalized for evaluation of the asymptotic approximation error. As an example, we discuss the approximation of the largest eigenvalue distribution of the Wishart matrix with a non-identity matrix parameter. The Bonferroni method is the tube method when the index set is a finite set. We provide the formula for the asymptotic approximation error for the Bonferroni method when the variance is not constant. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：In this paper, we develop a new procedure for estimating the parameters of a model by combining Zhang's (2019) recent Gaussian estimator and the minimum density power divergence estimators of Basu et al. (1998). The proposed estimator is called the Minimum Density Power Divergence Gaussian Estimator (MDPDGE). The consistency and asymptotic normality of the MDPDGE are proved. The MDPDGE is applied to some classical univariate distributions and it is also investigated for the family of elliptically contoured distributions. A numerical study illustrates the robustness of the proposed estimator. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：In many scientific areas the observations are collected with measurement errors. We are interested in measuring and testing independence between random vectors which are subject to measurement errors. We modify the weight functions in the classic distance covariance such that, the modified distance covariance between the random vectors of primary interest is the same as its classic version between the surrogate random vectors, which is referred to as the invariance law in the present context. The presence of measurement errors may substantially weaken the degree of nonlinear dependence. An immediate issue arises: The classic distance correlation between the surrogate vectors cannot reach one even if the two random vectors of primary interest are exactly linearly dependent. To address this issue, we propose to estimate the distance variance using repeated measurements. We study the asymptotic properties of the modified distance correlation thoroughly. In addition, we demonstrate its finite-sample performance through extensive simulations and a real-world application. (C) 2021 Elsevier Inc. All rights reserved.
NSTL
Elsevier