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Mathematical Methods Of Statistics

High dimension, low sample size data are emerging in various areas of science. We find a common structure underlying many such data sets by using a non-standard type of asymptotics: the dimension tends to ∞ while the sample size is fixed. Our analysis shows a tendency for the data to lie deterministically at the vertices of a regular simplex. Essentially all the randomness in the data appears only as a random rotation of this simplex. This geometric representation is used to obtain several new statistical insights. Copyright 2005 Royal Statistical Society.

Geometric representation of high dimension, low sample size data

A comparative analysis of high spatial resolution IKONOS and WorldView-2 imagery for mapping urban tree species

Cropping sequence diversification provides a systems approach to reduce yield variations and improve resilience to multiple environmental stresses. Yield advantages of more diverse crop rotations and their synergistic effects with reduced tillage are well documented, but few studies have quantified the impact of these management practices on yields and their stability when soil moisture is limiting or in excess. Using yield and weather data obtained from a 31-year long term rotation and tillage trial in Ontario, we tested whether crop rotation diversity is associated with greater yield stability when abnormal weather conditions occur. We used parametric and non-parametric approaches to quantify the impact of rotation diversity (monocrop, 2-crops, 3-crops without or with one or two legume cover crops) and tillage (conventional or reduced tillage) on yield probabilities and the benefits of crop diversity under different soil moisture and temperature scenarios. Although the magnitude of rotation benefits varied with crops, weather patterns and tillage, yield stability significantly increased when corn and soybean were integrated into more diverse rotations. Introducing small grains into short corn-soybean rotation was enough to provide substantial benefits on long-term soybean yields and their stability while the effects on corn were mostly associated with the temporal niche provided by small grains for underseeded red clover or alfalfa. Crop diversification strategies increased the probability of harnessing favorable growing conditions while decreasing the risk of crop failure. In hot and dry years, diversification of corn-soybean rotations and reduced tillage increased yield by 7% and 22% for corn and soybean respectively. Given the additional advantages associated with cropping system diversification, such a strategy provides a more comprehensive approach to lowering yield variability and improving the resilience of cropping systems to multiple environmental stresses. This could help to sustain future yield levels in challenging production environments.

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Increasing Crop Diversity Mitigates Weather Variations and Improves Yield Stability

We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables {X k } of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {X ij } j>i of zero mean and unit variance, scaling the eigenvalues by √n we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions γ H , γ M and γ T of unbounded support. The moments of γ H and γ T are the sum of volumes of solids related to Eulerian numbers, whereas γ M has a bounded smooth density given by the free convolution of the semicircle and normal densities. For symmetric Markov matrices generated by i.i.d. random variables {X iy } y>i of mean m and finite variance, scaling the eigenvalues by n we prove the almost sure, weak convergence of the spectral measures to the atomic measure at -m. If m = 0, and the fourth moment is finite, we prove that the spectral norm of M n scaled by √2n log n converges almost surely to 1.

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Spectral measure of large random Hankel, Markov and Toeplitz matrices

Preface. Glossary of Notation and Abbreviations. 1 Multivariate Normal and Related Distributions. 1.1 Random Vectors. 1.1.1 Mean Vector and Covariance Matrix. 1.1.2 Characteristic Function and Distribution. 1.2 Multivariate Normal Distribution. 1.2.1 Bivariate Normal Distribution. 1.2.2 Definition. 1.2.3 Some Properties. 1.3 Spherical and Elliptical Distributions. 1.4 Multivariate Cumulants. Problems. 2 Wishart Distribution. 2.1 Definition. 2.2 Some Basic Properties. 2.3 Functions of Wishart Matrices. 2.4 Cochran's Theorem. 2.5 Asymptotic Distributions. Problems. 3 Hotelling's T2 and LambdaStatistics. 3.1 Hotelling's T2 and LambdaStatistics. 3.1.1 Distribution of the T2Statistic. 3.1.2 Decomposition of T2 andD2. 3.2 Lambda-Statistic. 3.2.1 Motivation of Lambda Statistic. 3.2.2 Distribution of Lambda Statistic. 3.3 Test for Additional Information. 3.3.1 Decomposition of Lambda Statistic. Problems. 4 Correlation Coefficients. 4.1 Ordinary Correlation Coefficients. 4.1.1 Population Correlation. 4.1.2 Sample Correlation. 4.2 Multiple Correlation Coefficient. 4.2.1 Population Multiple Correlation. 4.2.2 Sample Multiple Correlation. 4.3 Partial Correlation. 4.3.1 Population Partial Correlation. 4.3.2 Sample Partial Correlation. 4.3.3 Covariance Selection Model. Problems. 5 Asymptotic Expansions for Multivariate BasicStatistics. 5.1 Edgeworth Expansion and its Validity. 5.2 The Sample Mean Vector and Covariance Matrix. 5.3 T2Statistic. 5.3.1 Outlines of Two Methods. 5.3.2 Multivariate t-Statistic. 5.3.3 Asymptotic Expansions. 5.4 Statistics with a Class of Moments. 5.4.1 Large-Sample Expansions. 5.4.2 High-Dimensional Expansions. 5.5 Perturbation Method. 5.6 Cornish-Fisher Expansions. 5.6.1 Expansion Formulas. 5.6.2 Validity of Cornish-Fisher Expansions. 5.7 Transformations for Improved Approximations. 5.8 Bootstrap Approximations. 5.9 High-Dimensional Approximations. 5.9.1 Limiting Spectral Distribution. 5.9.2 Central Limit Theorem. 5.9.3 Martingale Limit Theorem. 5.9.4 Geometric Representation. Problems. 6 MANOVA Models. 6.1 Multivariate One-Way Analysis of Variance. 6.2 Multivariate Two-Way Analysis of Variance. 6.3 MANOVA Tests. 6.3.1 Test Criteria. 6.3.2 Large-Sample Approximations. 6.3.3 Comparison of Powers. 6.3.4 High-Dimensional Approximations. 6.4 Approximations Under Nonnormality. 6.4.1 Asymptotic Expansions. 6.4.2 Bootstrap Tests. 6.5 Distributions of Characteristic Roots. 6.5.1 Exact Distributions. 6.5.2 Large-Sample Case. 6.5.3 High-Dimensional Case. 6.6 Tests for Dimensionality. 6.6.1 Three Test Criteria. 6.6.2 Large-Sample and High-Dimensional Asymptotics. 6.7 High-Dimensional Tests. Problems. 7 Multivariate Regression. 7.1 Multivariate Linear Regression Model. 7.2 Statistical Inference. 7.3 Selection of Variables. 7.3.1 Stepwise Procedure. 7.3.2 Cp Criterion. 7.3.3 AIC Criterion. 7.3.4 Numerical Example. 7.4 Principal Component Regression. 7.5 Selection of Response Variables. 7.6 General Linear Hypotheses and Confidence Intervals. 7.7 Penalized Regression Models. Problems. 8 Classical and High-Dimensional Tests for CovarianceMatrices. 8.1 Specified Covariance Matrix. 8.1.1 Likelihood Ratio Test and Moments. 8.1.2 Asymptotic Expansions. 8.1.3 High-Dimensional Tests. 8.2 Sphericity. 8.2.1 Likelihood Ratio Tests and Moments. 8.2.2 Asymptotic Expansions. 8.2.3 High-Dimensional Tests. 8.3 Intraclass Covariance Structure. 8.3.1 Likelihood Ratio Tests and Moments. 8.3.2 Asymptotic Expansions. 8.3.3 Numerical Accuracy. 8.4 Test for Independence. 8.4.1 Likelihood Ratio Tests and Moments. 8.4.2 Asymptotic Expansions. 8.4.3 High-Dimensional Tests. 8.5 Tests for Equality of Covariance Matrices. 8.5.1 Likelihood Ratio Test and Moments. 8.5.2 Asymptotic Expansions. 8.5.3 High-Dimensional Tests. Problems. 9 Discriminant Analysis. 9.1 Classification Rules for Known Distributions. 9.2 Sample Classification Rules for Normal Populations. 9.2.1 Two Normal Populations with S1 =S2. 9.2.2 Case of Several Normal Populations. 9.3 Probability of Misclassifications. 9.3.1 W-Rule. 9.3.2 Z-Rule. 9.3.3 High-Dimensional Asymptotic Results. 9.4 Canonical Discriminant Analysis. 9.4.1 Canonical Discriminant Method. 9.4.2 Test for Additional Information. 9.4.3 Selection of Variables. 9.4.4 Estimation of Dimensionality. 9.5 Regression Approach. 9.6 High-Dimensional Approach. 9.6.1 Penalized Discriminant Analysis. 9.6.2 Other Approaches. Problems. 10 Principal Component Analysis. 10.1 Definition of Principal Components. 10.2 Optimality of Principal Components. 10.3 Sample Principal Components. 10.4 MLEs of the Characteristic Roots and Vectors. 10.5 Distributions of the Characteristic Roots. 10.5.1 Exact Distribution. 10.5.2 Large-Sample Case. 10.5.3 High-Dimensional Case. 10.6 Model Selection Approach for Covariance Structures. 10.6.1 General Approach. 10.6.2 Models for Equality of the Smaller Roots. 10.6.3 Selecting a Subset of Original Variables. 10.7 Methods Related to Principal Components. 10.7.1 Fixed-Effect Principal Component Model. 10.7.2 Random-Effect Principal Components Model. Problems. 11 Canonical Correlation Analysis. 11.1 Definition of Population Canonical Correlations andVariables. 11.2 Sample Canonical Correlations. 11.3 Distributions of Canonical Correlations. 11.3.1 Distributional Reduction. 11.3.2 Large-Sample Asymptotic Distributions. 11.3.3 High-Dimensional Asymptotic Distributions. 11.3.4 Fisher's z-Transformation. 11.4 Inference for Dimensionality. 11.4.1 Test of Dimensionality. 11.4.2 Estimation of Dimensionality. 11.5 Selection of Variables. 11.5.1 Test for Redundancy. 11.5.2 Selection of Variables. Problems. 12 Growth Curve Analysis. 12.1 Growth Curve Model. 12.2 Statistical Inference: One Group. 12.2.1 Test for Adequacy. 12.2.2 Estimation and Test. 12.2.3 Confidence Intervals. 12.3 Statistical Methods: Several Groups. 12.4 Derivation of Statistical Inference. 12.4.1 A General Multivariate Linear Model. 12.4.2 Estimation. 12.4.3 LR Tests for General Linear Hypotheses. 12.4.4 Confidence Intervals. 12.5 Model Selection. 12.5.1 AIC and CAIC. 12.5.2 Derivation of CAIC. 12.5.3 Extended Growth Curve Model. Problems. 13 Approximation to the Scale-Mixted Distributions. 13.1 Introduction. 13.1.1 Simple Example: Student's t-Distribution. 13.1.2 Improving the Approximation. 13.2 Error Bounds Evaluated in Sup-Norm. 13.2.1 General Theory. 13.2.2 Scale-Mixed Normal. 13.2.3 Scale-Mixed Gamma. 13.3 Error Bounds Evaluated inL1-Norm. 13.3.1 Some Basic Results. 13.3.2 Scale-Mixed Normal Density. 13.3.3 Scale-Mixed Gamma Density. 13.3.4 Scale-Mixed Chi-square Density. 13.4 Multivariate Scale Mixtures. 13.4.1 General Theory. 13.4.2 Normal Case. 13.4.3 Gamma Case. Problems. 14 Approximation to Some Related Distributions. 14.1 Location and Scale Mixtures. 14.2 Maximum of Multivariate Variables. 14.2.1 Distribution of the Maximum Component of a MultivariateVariable. 14.2.2 Multivariate t-Distribution. 14.2.3 Multivariate F-Distribution. 14.3 Scale Mixtures of the F-Distribution. 14.4 Non-Uniform Error Bounds. 14.5 Method of Characteristic Functions. Problems. 15 Error Bounds for Approximations of MultivariateTests. 15.1 Multivariate Scale Mixture and MANOVA Tests. 15.2 A Function of Multivariate Scale Mixture. 15.3 Hotelling's T²0Statistic. 15.4 Wilk's Lambda Distribution. 15.4.1 Univariate Case. 15.4.2 Multivariate Case. Problems. 16 Error Bounds for Approximations to Some OtherStatistics. 16.1 Linear Discriminant Function. 16.1.1 Representation as Location and Scale Mixture. 16.1.2 Large-Sample Approximations. 16.1.3 High-Dimensional Approximations. 16.1.4 Some Related Topics. 16.2 Profile Analysis. 16.2.1 Parallelism Model and MLE. 16.2.2 Distributions of . 16.2.3 Confidence Interval for . 16.3 Estimators in the Growth Curve Model. 16.3.1 Error Bounds. 16.3.2 Distribution of the Bilinear Form. 16.4 Generalized Least Squares Estimators. Problems. Appendix. A.1 Some Results on Matrices. A.1.1 Determinants and Inverse Matrices. A.1.2 Characteristic Roots and Vectors. A.1.3 Matrix Factorizations. A.1.4 Idempotent Matrices. A.2 Inequalities and Max-Min Problems. A.3 Jacobians of Transformations. Bibliography. Index.

Multivariate Statistics: High-Dimensional and Large-Sample Approximations

On the accuracy of normal approximation

Let $X_{1},\ldots,X_{n}$ be i.i.d. sample in $\mathbb{R}^{p}$ with zero mean and the covariance matrix $\mathbf{\Sigma}$. The problem of recovering the projector onto an eigenspace of $\mathbf{\Sigma}$ from these observations naturally arises in many applications. Recent technique from [Koltchinskii, Lounici, 2015] helps to study the asymptotic distribution of the distance in the Frobenius norm $\| \mathbf{P}_r - \widehat{\mathbf{P}}_r \|_{2}$ between the true projector $\mathbf{P}_r$ on the subspace of the $r$-th eigenvalue and its empirical counterpart $\widehat{\mathbf{P}}_r$ in terms of the effective rank of $\mathbf{\Sigma}$. This paper offers a bootstrap procedure for building sharp confidence sets for the true projector $\mathbf{P}_r$ from the given data. This procedure does not rely on the asymptotic distribution of $\| \mathbf{P}_r - \widehat{\mathbf{P}}_r \|_{2}$ and its moments. It could be applied for small or moderate sample size $n$ and large dimension $p$. The main result states the validity of the proposed procedure for finite samples with an explicit error bound for the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high-dimensional spaces. Numeric results confirm a good performance of the method in realistic examples.

Bootstrap confidence sets for spectral projectors of sample covariance

This is a companion book to Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions by A.A. Borovkov and K.A. Borovkov. Its self-contained systematic exposition provides a highly useful resource for academic researchers and professionals interested in applications of probability in statistics, ruin theory, and queuing theory. The large deviation principle for random walks was first established by the author in 1967, under the restrictive condition that the distribution tails decay faster than exponentially. (A close assertion was proved by S.R.S. Varadhan in 1966, but only in a rather special case.) Since then, the principle has always been treated in the literature only under this condition. Recently, the author jointly with A.A. Mogul'skii removed this restriction, finding a natural metric for which the large deviation principle for random walks holds without any conditions. This new version is presented in the book, as well as a new approach to studying large deviations in boundary crossing problems. Many results presented in the book, obtained by the author himself or jointly with co-authors, are appearing in a monograph for the first time.

Asymptotic Analysis of Random Walks: Light-Tailed Distributions

We review the results about the accuracy of approximations for distributions of functionals of sums of independent random elements with values in a Hilbert space. Mainly we consider recent results for quadratic and almost quadratic forms motivated by asymptotic problems in mathematical statistics. Some of the results are optimal and could not be further improved without additional conditions.

Vladimir V. Ulyanov

Papers

Multivariate Statistics: High-Dimensional and Large-Sample Approximations

On the accuracy of normal approximation

Bootstrap confidence sets for spectral projectors of sample covariance

Asymptotic Analysis of Random Walks: Light-Tailed Distributions

Some Approximation Problems in Statistics and Probability