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Central Limit Theorems for Classical Likelihood Ratio Tests for HighDimensional Normal Distributions
"... For random samples of size n obtained from pvariate normal distributions, we consider the classical likelihood ratio tests (LRT) for their means and covariance matrices in the highdimensional setting. These test statistics have been extensively studied in multivariate analysis and their limiting d ..."
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Cited by 11 (4 self)
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For random samples of size n obtained from pvariate normal distributions, we consider the classical likelihood ratio tests (LRT) for their means and covariance matrices in the highdimensional setting. These test statistics have been extensively studied in multivariate analysis and their limiting distributions under the null hypothesis were proved to be chisquare distributions as n goes to infinity and p remains fixed. In this paper, we consider the highdimensional case where both p and n go to infinity with p/n → y ∈ (0, 1]. We prove that the likelihood ratio test statistics under this assumption will converge in distribution to normal distributions with explicit means and variances. We perform the simulation study to show that the likelihood ratio tests using our central limit theorems outperform those using the traditional chisquare approximations for analyzing highdimensional data.
Optimal tests for homogeneity of covariance, scale, and shape
 J. Multivariate Anal
, 2008
"... The assumption of homogeneity of covariance matrices is the fundamental prerequisite of a number of classical procedures in multivariate analysis. Despite its importance and long history, however, this problem so far has not been completely settled beyond the traditional and highly unrealistic cont ..."
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Cited by 7 (4 self)
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The assumption of homogeneity of covariance matrices is the fundamental prerequisite of a number of classical procedures in multivariate analysis. Despite its importance and long history, however, this problem so far has not been completely settled beyond the traditional and highly unrealistic context of multivariate Gaussian models. And the modified likelihood ratio tests (MLRT) that are used in everyday practice are known to be highly sensitive to violations of Gaussian assumptions. In this paper, we provide a complete and systematic study of the problem, and propose test statistics which, while preserving the optimality features of the MLRT under multinormal assumptions, remain valid under unspecified elliptical densities with finite fourthorder moments. As a first step, the Le Cam LAN approach is used for deriving locally and asymptotically optimal testing procedures φ (n) f for any specified mtuple of radial densities f = (f1,..., fm). Combined with an estimation of the m densities f1,..., fm, these procedures can be used to construct adaptive tests for the problem. Adaptive tests however typically require very large samples, and pseudoGaussian tests—namely, tests that are locally and asymptotically optimal at Gaussian densities while remaining valid under a much broader class of distributions—in general are preferable. We therefore construct two pseudoGaussian modifications of the Gaussian version φ (n) N of the optimal test φ (n) f. The first one, φ
Inferences on a normal covariance matrix and generalized variance with monotone missing data
 J. Multivariate Anal
, 2001
"... The problems of testing a normal covariance matrix and an interval estimation of generalized variance when the data are missing from subsets of components are considered. The likelihood ratio test statistic for testing the covariance matrix is equal to a specified matrix, and its asymptotic null dis ..."
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Cited by 2 (0 self)
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The problems of testing a normal covariance matrix and an interval estimation of generalized variance when the data are missing from subsets of components are considered. The likelihood ratio test statistic for testing the covariance matrix is equal to a specified matrix, and its asymptotic null distribution is derived when the data matrix is of a monotone pattern. The validity of the asymptotic null distribution and power analysis are performed using simulation. The problem of testing the normal mean vector and a covariance matrix equal to a given vector and matrix is also addressed. Further, an approximate confidence interval for the generalized variance is given. Numerical studies show that the proposed interval estimation procedure is satisfactory even for small samples. The results are illustrated using simulated data. 2001 Academic Press AMS 1991 subject classifications: 62F25; 62H99. Key words and phrases: generalized variance; likelihood ratio test; missing data; monotone patterns power; Satterthwaite approximation. 1.
Exact Inference with Monotone Incomplete Multivariate Normal Data
, 2006
"... We consider problems in finitesample inference with twostep, monotone incomplete data drawn from Nd(µ, Σ), a multivariate normal population with mean µ and covariance matrix Σ. We derive stochastic representations for the distributions of ̂µ and ̂ Σ, the maximum likelihood estimators of µ and Σ, r ..."
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We consider problems in finitesample inference with twostep, monotone incomplete data drawn from Nd(µ, Σ), a multivariate normal population with mean µ and covariance matrix Σ. We derive stochastic representations for the distributions of ̂µ and ̂ Σ, the maximum likelihood estimators of µ and Σ, respectively. Under the assumption that Σ is blockdiagonal when partitioned according to the twostep pattern, we derive the distributions of the diagonal blocks of ̂ Σ and of the estimated regression matrix, ̂Σ12 ̂ Σ −1 22. We obtain a representation for ̂ Σ in terms of independent matrices, and then derive the exact density function and saddlepoint approximations thereof for ̂ Σ and its partial Iwasawa coordinates. We obtain ellipsoidal confidence regions for µ based on T 2, a generalization of Hotelling’s T 2statistic; we derive probability inequalities for, and the asymptotic distribution of, T 2 under various assumptions on the sizes of the complete and incomplete samples, and we apply these results to construct confidence regions for
BY
, 2011
"... extremely interested in probability theory and the challenges and opportunities that this topic presents in statistics, and many other areas of research. During the time that I have been constructing my thesis, I have had the chance to reconsider, revise, and refine many of the original concepts and ..."
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extremely interested in probability theory and the challenges and opportunities that this topic presents in statistics, and many other areas of research. During the time that I have been constructing my thesis, I have had the chance to reconsider, revise, and refine many of the original concepts and methodologies foundational to my research. I have also had the opportunity to explore some of these concepts in applied settings, which has helped me to integrate my theoretical exploration in ways that have proven useful and practical. The combined research and writing process has been a rich and enlightening one for me, and over time, I have grown and evolved both in my thinking and in my approach to research. As every doctoral student can attest, completion of a dissertation cannot be accomplished in isolation. It involves the support and contribution of many people, both inside and outside the department. I would first like to thank my advisor, Professor Tiefeng Jiang. I am deeply indebted to him for his support and commitment, his enthusiastic guidance, and his insight throughout the research and writing process. He could not have been more generous with his time and effort in directing me toward the
A New Scheme for Monitoring Multvariate Process Dispersion
, 2009
"... Construction of control charts for multivariate process dispersion is not as straightforward as for the process mean. Because of the complexity of out of control scenarios, a general method is not available. In this dissertation, we consider the problem of monitoring multivariate dispersion from tw ..."
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Construction of control charts for multivariate process dispersion is not as straightforward as for the process mean. Because of the complexity of out of control scenarios, a general method is not available. In this dissertation, we consider the problem of monitoring multivariate dispersion from two perspectives. First, we derive asymptotic approximations to the power of Nagao’s test for the equality of a normal dispersion matrix to a given constant matrix under local and fixed alternatives. Second, we propose various unequally weighted sum of squares estimators for the dispersion matrix, particularly with exponential weights. The new estimators give more weights to more recent observations and are not exactly Wishart distributed. Satterthwaite’s method is used to approximate the distribution of the new estimators. By combining these two techniques based on exponentially weighted sums of squares and Nagao’s test, we are able to propose a new control scheme MTNT, which is easy to implement. The control limits are easily calculated since they only depend on the dimension of the process and the desired in control average run length. Our simulations show that compared with schemes based on the likelihood ratio test and the sample generalized variance, MTNT has the shortest out of control average run length for a variety of out of control scenarios, particularly when process variances increase.