Verfügbarkeit: Multivariate statistics

Multivariate statistics: high-dimensional and large-sample approximations

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Bibliographische Detailangaben
Hauptverfasser:	Fujikoshi, Yasunori 1942- (VerfasserIn), Ulʹjanov, Vladimir Vladimirovič 1953- (VerfasserIn), Shimizu, Ryoichi 1931- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Hoboken, NJ Wiley 2010
Schriftenreihe:	Wiley series in probability and statistics
Schlagworte:	Multivariate analysis Approximation theory Multivariate Analyse Lehrbuch
Online-Zugang:	Inhaltsverzeichnis Klappentext
Beschreibung:	XVIII, 533 S.
ISBN:	9780470411698

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Datensatz im Suchindex

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adam_text	Contents Preface .................................................................xiii Glossary of Notation and Abbreviations ...........................xvii 1 Multivariate Normal and Related Distributions ......................1 1.1 Random Vectors .......................................................1 1.1.1 Mean Vector and Covariance Matrix ............................1 1.1.2 Characteristic Function and Distribution .......................5 1.2 Multivariate Normal Distribution ......................................6 1.2.1 Bivariate Normal Distribution ..................................6 1.2.2 Definition ......................................................8 1.2.3 Some Properties ..............................................10 1.3 Spherical and Elliptical Distributions .................................15 1.4 Multivariate Cumulants ..............................................19 Problems ............................................................24 2 Wishart Distribution ..................................................29 2.1 Definition ............................................................29 2.2 Some Basic Properties ................................................32 2.3 Functions of Wishart Matrices ........................................36 2.4 Cochran s Theorem ..................................................39 2.5 Asymptotic Distributions .............................................40 Problems ............................................................43 3 Hotelling s T2 and Lambda Statistics ................................47 3.1 Hotelling s T2 and Lambda Statistics .................................47 3.1.1 Distribution of the T2 Statistic ................................47 3.1.2 Decomposition of T2 and D2 ..................................49 3.2 Lambda Statistic .....................................................53 3.2.1 Motivation of the Lambda Statistic ...........................53 3.2.2 Distribution of the Lambda Statistic ..........................55 3.3 Test for Additional Information ......................................58 vi Contents 3.3.1 Decomposition of the Lambda Statistic ........................61 Problems ............................................................64 4 Correlation Coefficients ................................................69 4.1 Ordinary Correlation Coefficients .....................................69 4.1.1 Population Correlation ........................................69 4.1.2 Sample Correlation ...........................................71 4.2 Multiple Correlation Coefficient ......................................75 4.2.1 Population Multiple Correlation ...............................75 4.2.2 Sample Multiple Correlation ..................................77 4.3 Partial Correlation ...................................................80 4.3.1 Population Partial Correlation ................................80 4.3.2 Sample Partial Correlation ....................................82 4.3.3 Covariance Selection Model ...................................83 Problems ............................................................87 5 Asymptotic Expansions for Multivariate Basic Statistics ...........91 5.1 Edgeworth Expansion and its Validity ................................91 5.2 Sample Mean Vector and Covariance Matrix ..........................98 5.3 T2 Statistic .........................................................104 5.3.1 Outlines of Two Methods ....................................104 5.3.2 Multivariate t-Statistic .......................................107 5.3.3 Asymptotic Expansions ......................................109 5.4 Statistics with a Class of Moments .................................. Ill 5.4.1 Large-Sample Expansions .................................... Ill 5.4.2 High-Dimensional Expansions ................................117 5.5 Perturbation Method ................................................120 5.6 Cornish-Fisher Expansions ..........................................125 5.6.1 Expansion Formulas .........................................125 5.6.2 Validity of Cornish-Fisher Expansions .......................129 5.7 Transformations for Improved Approximations ......................132 5.8 Bootstrap Approximations ..........................................135 5.9 High-Dimensional Approximations ..................................138 5.9.1 Limiting Spectral Distribution ...............................138 5.9.2 Central Limit Theorem ......................................140 5.9.3 Martingale Limit Theorem ...................................143 5.9.4 Geometric Representation ....................................144 Problems ...........................................................145 6 MÁNOVA Models .....................................................149 6.1 Multivariate One-Way Analysis of Variance ..........................149 6.2 Multivariate Two-Way Analysis of Variance ..........................152 6.3 MÁNOVA Tests ....................................................157 Contents vii 6.3.1 Test Criteria .................................................157 6.3.2 Large-Sample Approximations ...............................158 6.3.3 Comparison of Powers .......................................159 6.3.4 High-Dimensional Approximations ...........................161 6.4 Approximations Under Nonnormality ................................163 6.4.1 Asymptotic Expansions ......................................163 6.4.2 Bootstrap Tests ..............................................167 6.5 Distributions of Characteristic Roots ................................170 6.5.1 Exact Distributions ..........................................170 6.5.2 Large-Sample Case ...........................................172 6.5.3 High-Dimensional Case ......................................174 6.6 Tests for Dimensionality ............................................176 6.6.1 Three Test Criteria ..........................................176 6.6.2 Large-Sample and High-Dimensional Asymptotics ............178 6.7 High-Dimensional Tests .............................................181 Problems ...........................................................183 7 Multivariate Regression ..............................................187 7.1 Multivariate Linear Regression Model ...............................187 7.2 Statistical Inference .................................................189 7.3 Selection of Variables ...............................................194 7.3.1 Stepwise Procedure ..........................................194 7.3.2 Cp Criterion .................................................196 7.3.3 AIC Criterion ................................................200 7.3.4 Numerical Example ..........................................202 7.4 Principal Component Regression ....................................203 7.5 Selection of Response Variables .....................................206 7.6 General Linear Hypotheses and Confidence Intervals ................209 7.7 Penalized Regression Models ........................................213 Problems ...........................................................213 8 Classical and High-Dimensional Tests for Covariance Matrices ..............................................................219 8.1 Specified Covariance Matrix .........................................219 8.1.1 Likelihood Ratio Test and Moments ..........................219 8.1.2 Asymptotic Expansions ......................................221 8.1.3 High-Dimensional Tests ......................................225 8.2 Sphericity ...........................................................227 8.2.1 Likelihood Ratio Tests and Moments .........................227 8.2.2 Asymptotic Expansions ......................................228 8.2.3 High-Dimensional Tests ......................................230 8.3 Intraclass Covariance Structure .....................................231 viii Contents 8.3.1 Likelihood Ratio Tests and Moments .........................231 8.3.2 Asymptotic Expansions ......................................233 8.3.3 Numerical Accuracy .........................................235 8.4 Test for Independence ...............................................236 8.4.1 Likelihood Ratio Tests and Moments .........................236 8.4.2 Asymptotic Expansions ......................................238 8.4.3 High-Dimensional Tests ......................................239 8.5 Tests for Equality of Covariance Matrices ...........................241 8.5.1 Likelihood Ratio Test and Moments ..........................241 8.5.2 Asymptotic Expansions ......................................243 8.5.3 High-Dimensional Tests ......................................244 Problems ...........................................................245 9 Discriminant Analysis ................................................249 9.1 Classification Rules for Known Distributions ........................249 9.2 Sample Classification Rules for Normal Populations .................256 9.2.1 Two Normal Populations with Σι = Σ2 ...................... 256 9.2.2 Case of Several Normal Populations ..........................258 9.3 Probability of Misclassifications .....................................258 9.3.1 W-Rule ......................................................259 9.3.2 Z-Rule .......................................................261 9.3.3 High-Dimensional Asymptotic Results ........................263 9.4 Canonical Discriminant Analysis ....................................265 9.4.1 Canonical discriminant Method ..............................265 9.4.2 Test for Additional Information ..............................267 9.4.3 Selection of Variables ........................................270 9.4.4 Estimation of Dimensionality ................................273 9.5 Regression Approach ................................................276 9.6 High-Dimensional Approach .........................................278 9.6.1 Penalized Discriminant Analysis .............................278 9.6.2 Other Approaches ...........................................278 Problems ...........................................................280 10 Principal Component Analysis ......................................283 10.1 Definition of Principal Components .................................283 10.2 Optimality of Principal Components ................................286 10.3 Sample Principal Components ......................................288 10.4 MLEs of the Characteristic Roots and Vectors ......................291 10.5 Distributions of the Characteristic Roots ............................292 10.5.1 Exact Distribution ..........................................293 10.5.2 Large-sample Case ..........................................294 10.5.3 High-dimensional Case ......................................301 Contents ix 10.6 Model Selection Approach for Covariance Structures ................302 10.6.1 General Approach ..........................................302 10.6.2 Models for Equality of the Smaller Roots ...................305 10.6.3 Selecting a Subset of Original Variables .....................306 10.7 Methods Related to Principal Components ..........................308 10.7.1 Fixed-Effect Principal Component Model ...................308 10.7.2 Random-Effect Principal Components Model ................310 Problems ..........................................................311 11 Canonical Correlation Analysis ......................................317 11.1 Definition of Population Canonical Correlations and Variables .........................................................317 11.2 Sample Canonical Correlations .....................................322 11.3 Distributions of Canonical Correlations .............................324 11.3.1 Distributional Reduction ...................................324 11.3.2 Large-Sample Asymptotic Distribuitons ....................326 11.3.3 High-Dimensional Asymptotic Distributions ................327 11.3.4 Fisher s z-Transformation ..................................333 11.4 Inference for Dimensionality ........................................335 11.4.1 Test of Dimensionality .....................................335 11.4.2 Estimation of Dimensionality ..............................337 11.5 Selection of Variables ...............................................338 11.5.1 Test for Redundancy .......................................338 11.5.2 Selection of Variables ......................................342 Problems ..........................................................345 12 Growth Curve Analysis ...............................................349 12.1 Growth Curve Model ...............................................349 12.2 Statistical Inference: One Group ....................................352 12.2.1 Test for Adequacy .........................................352 12.2.2 Estimation and Test .......................................354 12.2.3 Confidence Intervals .......................................357 12.3 Statistical Methods: Several Groups ................................359 12.4 Derivation of Statistical Inference ...................................365 12.4.1 General Multivariate Linear Model .........................365 12.4.2 Estimation .................................................366 12.4.3 LR Tests for General Linear Hypotheses ...................368 12.4.4 Confidence Intervals .......................................369 12.5 Model Selection ....................................................370 12.5.1 AIC and CAIC ............................................370 12.5.2 Derivation of CAIC ........................................371 12.5.3 Extended Growth Curve Model ............................373 x Contents Problems..........................................................376 13 Approximation to the Scale- Mixted Distributions .................379 13.1 Introduction ........................................................379 13.1.1 Simple Example: Student s t-Distribution ..................379 13.1.2 Improving the Approximation ..............................381 13.2 Error Bounds evaluated in sup-Norm ...............................384 13.2.1 General Theory ............................................384 13.2.2 Scale-Mixed Normal .......................................388 13.2.3 Scale-Mixed Gamma .......................................390 13.3 Error Bounds evaluated in Li-Norm ................................395 13.3.1 Some Basic Results ........................................395 13.3.2 Scale-Mixed Normal Density ...............................397 13.3.3 Scale-Mixed Gamma Density ...............................399 13.3.4 Scale-Mixed Chi-square Density ............................402 13.4 Multivariate Scale Mixtures .........................................404 13.4.1 General Theory ............................................404 13.4.2 Normal Case ...............................................410 13.4.3 Gamma Case ..............................................415 Problems ..........................................................418 14 Approximation to Some Related Distributions ....................423 14.1 Location and Scale Mixtures ........................................423 14.2 Maximum of Multivariate Variables .................................426 14.2.1 Distribution of the Maximum Component of a Multi¬ variate Variable ..............................................426 14.2.2 Multivariate t-Distribution .................................427 14.2.3 Multivariate F-Distribution ................................429 14.3 Scale Mixtures of the F-Distribution ................................430 14.4 Nonuniform Error Bounds ..........................................433 14.5 Method of Characteristic Functions .................................436 Problems ..........................................................439 15 Error Bounds for Approximations of Multivariate Tests ..........441 15.1 Multivariate Scale Mixture and MÁNOVA Tests ....................441 15.2 Function of a Multivariate Scale Mixture ...........................443 15.3 Hotelling s To Statistic .............................................445 15.4 Wilk s Lambda Distribution ........................................448 15.4.1 Univariate Case ............................................448 15.4.2 Multivariate Case ..........................................456 Problems ..........................................................465 16 Error Bounds for Approximations to Some Other Statistics .....467 Contents xi 16.1 Lineai Discriminant Function .......................................467 16.1.1 Representation as a Location and Scale Mixture ............467 16.1.2 Large-Sample Approximations .............................472 16.1.3 High-Dimensional Approximations .........................474 16.1.4 Some Related Topics .......................................476 16.2 Profile Analysis ....................................................479 16.2.1 Parallelism Model and MLE ...............................479 16.2.2 Distributions of 7 ..........................................481 16.2.3 Confidence Interval for 7...................................486 16.3 Estimators in the Growth Curve Model .............................487 16.3.1 Error Bounds ..............................................487 16.3.2 Distribution of the Bilinear Form ..........................488 16.4 Generalized Least Squares Estimators ..............................490 Problems ..........................................................492 Appendix ..............................................................495 A.I Some Results on Matrices ..........................................495 A. 1.1 Determinants and Inverse Matrices .........................495 A.I. 2 Characteristic Roots and Vectors ...........................496 A. 1.3 Matrix Factorizations .......................................497 A. 1.4 Idempotent Matrices .......................................500 A.2 Inequalities and Max-Min Problems ................................502 A.3 Jacobians of Transformations .......................................508 Bibliography ..........................................................513 Index ..................................................................527 A comprehensive examination of high-dimensional analysis of multivariate methods and their real-world applications ultivciriate Statistics: High-Dimensional and Large-Sample Approximations is the first book of its kind to explore how classical multivariate methods can be revised and used in place of conventional statistical tools. Written by prominent researchers in the field, the book focuses on high-dimensional and large-scale approximations and details the many basic multivariate methods used to achieve high levels of accuracy. The authors begin with a fundamental presentation of the basic tools and exact distribu¬ tional results of multivariate statistics, and, in addition, the derivations of most distributional results are provided. Statistical methods for high-dimensional data, such as curve data, spectra, images, and DNA microarrays, are discussed. Bootstrap approximations from a methodological point of view, theoretical accuracies in MÁNOVA tests, and model selection criteria are also presented. Subsequent chapters feature additional topical coverage including: High-dimensional approximations of various statistics High-dimensional statistical methods Approximations with computable error bound • Selection of variables based on model selection approach 1 Statistics with error bounds and their appearance in discriminant analysis, growth curve models, generalized linear models, profile analysis, and multiple comparison Each chapter provides real-world applications and thorough analyses of the real data. In addition, approximation formulas found throughout the book are a useful tool for both practical and theoretical statisticians, and basic results on exact distributions in multivariate analysis are included in a comprehensive, yet accessible, format. Wultivariate Statistics is an excellent book for courses on probability theory in statistics at the graduate level. It is also an essential reference for both practical and theoretical statisticians who are interested in multivariate analysis and who would benefit from learning the applications of analytical probabilistic methods in statistics. is Professor Emeritus at Hiroshima University (Japan) and Visiting Professor in the Department of Mathematics at Chuo University (Japan). He has authored over 150 journal articles in the area of multivariate analysis. is Professor in the Department of Mathematical Statistics at Moscow State University (Russia) and is the author of nearly fifty journal articles in his areas of research interest, which include weak limit theorems, probability measures on topological spaces, and Gaussian processes. is Professor Emeritus at the Institute of Statistical Mathe¬ matics (Japan) and is the author of numerous journal articles on probability distributions. nstics eNewsletter at ®WILEY wilev.com ISBN cl7ä-a-47D-411bc1-a 90000 9 780470Ч 11698
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spelling	Fujikoshi, Yasunori 1942- Verfasser (DE-588)140791671 aut Multivariate statistics high-dimensional and large-sample approximations Yasunori Fujikoshi ; Vladimir V. Ulyanov ; Ryoichi Shimizu Hoboken, NJ Wiley 2010 XVIII, 533 S. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics Multivariate analysis Approximation theory Multivariate Analyse (DE-588)4040708-1 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Multivariate Analyse (DE-588)4040708-1 s DE-604 Ulʹjanov, Vladimir Vladimirovič 1953- Verfasser (DE-588)140792015 aut Shimizu, Ryoichi 1931- Verfasser (DE-588)140792392 aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628506&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628506&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext
spellingShingle	Fujikoshi, Yasunori 1942- Ulʹjanov, Vladimir Vladimirovič 1953- Shimizu, Ryoichi 1931- Multivariate statistics high-dimensional and large-sample approximations Multivariate analysis Approximation theory Multivariate Analyse (DE-588)4040708-1 gnd
subject_GND	(DE-588)4040708-1 (DE-588)4123623-3
title	Multivariate statistics high-dimensional and large-sample approximations
title_auth	Multivariate statistics high-dimensional and large-sample approximations
title_exact_search	Multivariate statistics high-dimensional and large-sample approximations
title_full	Multivariate statistics high-dimensional and large-sample approximations Yasunori Fujikoshi ; Vladimir V. Ulyanov ; Ryoichi Shimizu
title_fullStr	Multivariate statistics high-dimensional and large-sample approximations Yasunori Fujikoshi ; Vladimir V. Ulyanov ; Ryoichi Shimizu
title_full_unstemmed	Multivariate statistics high-dimensional and large-sample approximations Yasunori Fujikoshi ; Vladimir V. Ulyanov ; Ryoichi Shimizu
title_short	Multivariate statistics
title_sort	multivariate statistics high dimensional and large sample approximations
title_sub	high-dimensional and large-sample approximations
topic	Multivariate analysis Approximation theory Multivariate Analyse (DE-588)4040708-1 gnd
topic_facet	Multivariate analysis Approximation theory Multivariate Analyse Lehrbuch
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628506&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628506&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT fujikoshiyasunori multivariatestatisticshighdimensionalandlargesampleapproximations AT ulʹjanovvladimirvladimirovic multivariatestatisticshighdimensionalandlargesampleapproximations AT shimizuryoichi multivariatestatisticshighdimensionalandlargesampleapproximations

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