A timely and applied approach to the newly discovered methods and applications of U-statisticsBuilt on years of collaborative research and academic experience, Modern Applied U-Statistics successfully presents a thorough introduction to the theory of U-statistics using in-depth examples and applications that address contemporary areas of study including biomedical and psychosocial research. Utilizing a "learn by example" approach, this book provides an accessible, yet in-depth, treatment of U-statistics, as well as addresses key concepts in asymptotic theory by integrating translational and cross-disciplinary research.The authors begin with an introduction of the essential and theoretical foundations of U-statistics such as the notion of convergence in probability and distribution, basic convergence results, stochastic Os, inference theory, generalized estimating equations, as well as the definition and asymptotic properties of U-statistics. With an emphasis on nonparametric applications when and where applicable, the authors then build upon this established foundation in order to equip readers with the knowledge needed to understand the modern-day extensions of U-statistics that are explored in subsequent chapters. Additional topical coverage includes:Longitudinal data modeling with missing dataParametric and distribution-free mixed-effect and structural equation modelsA new multi-response based regression framework for non-parametric statistics such as the product moment correlation, Kendall's tau, and Mann-Whitney-Wilcoxon rank testsA new class of U-statistic-based estimating equations (UBEE) for dependent responsesMotivating examples, in-depth illustrations of statistical and model-building concepts, and an extensive discussion of longitudinal study designs strengthen the real-world utility and comprehension of this book. An accompanying Web site features SAS? and S-Plus? program codes, software applications, and additional study data. Modern Applied U-Statistics accommodates second- and third-year students of biostatistics at the graduate level and also serves as an excellent self-study for practitioners in the fields of bioinformatics and psychosocial research.
Author(s): Jeanne Kowalski, Xin M. Tu
Series: Wiley Series in Probability and Statistics
Edition: 1
Publisher: Wiley
Year: 2007
Language: English
Pages: 402
Modern Applied U-Statistics......Page 3
Contents......Page 7
Preface......Page 11
1 Preliminaries......Page 15
1.1.1 The Linear Regression Model......Page 16
1.1.2 The Product–Moment Correlation......Page 19
1.1.3 The Rank-Based Mann–Whitney–Wilcoxon Test......Page 21
1.2.1 Measurable Space......Page 23
1.2.2 Measure Space......Page 26
1.3 Measurable Function and Integration......Page 28
1.3.1 Measurable Functions......Page 29
1.3.2 Convergence of Sequence of Measurable Functions......Page 30
1.3.3 Integration of Measurable Functions......Page 32
1.3.4 Integration of Sequences of Measurable Functions......Page 35
1.4.1 Probability Space......Page 38
1.4.2 Random Variables......Page 40
1.4.3 Random Vectors......Page 41
1.5.1 Distribution Function......Page 44
1.5.2 Joint Distribution of Random Vectors......Page 47
1.5.3 Expectation......Page 50
1.5.4 Conditional Expectation......Page 52
1.6.1 Modes of Convergence......Page 55
1.6.2 Convergence of Sequence of I.I.D. Random Variables......Page 57
1.6.3 Rate of Convergence of Random Sequence......Page 58
1.6.4 Stochastic op (.) and Op (.)......Page 62
1.7.1 Convergence of Functions of Random Variables......Page 64
1.7.2 Convergence of Functions of Random Vectors......Page 69
1.8 Exercises......Page 72
2 Models for Cross-Sectional Data......Page 77
2.1 Parametric Regression Models......Page 78
2.1.1 Linear Regression Model......Page 79
2.1.2 Inference for Linear Models......Page 82
2.1.3 General Linear Hypothesis......Page 95
2.1.4 Generalized Linear Models......Page 101
2.1.5 Inference for Generalized Linear Models......Page 115
2.2 Distribution-Free (Semiparametric) Models......Page 118
2.2.1 Distribution-Free Generalized Linear Models......Page 119
2.2.2 Inference for Generalized Linear Models......Page 121
2.3 Exercises......Page 129
3 Univariate U-Statistics......Page 133
3.1 U-Statistics and Associated Models......Page 134
3.1.1 One Sample U-Statistics......Page 135
3.1.2 Two-Sample and General K Sample U-Statistics......Page 145
3.1.3 Representation of U-Statistic by Order Statistic......Page 151
3.1.4 Martingale Structure of U-Statistic......Page 158
3.2 Inference for U-Statistics......Page 164
3.2.1 Projection of U-statistic......Page 166
3.2.2 Asymptotic Distribution of One-Group U-Statistic......Page 171
3.2.3 Asymptotic Distribution of K-Group U-Statistic......Page 178
3.3 Exercises......Page 184
4 Models for Clustered Data......Page 189
4.1 Longitudinal versus Cross-Sectional Designs......Page 190
4.2.1 Multivariate Normal Distribution Based Models......Page 193
4.2.2 Linear Mixed-Effects Model......Page 198
4.2.3 Generalized Linear Mixed-Effects Models......Page 205
4.2.4 Maximum Likelihood Inference......Page 207
4.3.1 Distribution-Free Models for Longitudinal Data......Page 212
4.3.2 Inference for Distribution-Free Models......Page 215
4.4 Missing Data......Page 223
4.4.1 Inference for Parametric Models......Page 225
4.4.2 Inference for Distribution-Free Models......Page 228
4.5 GEE II for Modeling Mean and Variance......Page 237
4.6.1 Path Diagrams and Models......Page 244
4.6.2 Maximum Likelihood Inference......Page 251
4.6.3 GEE-Based Inference......Page 255
4.7 Exercises......Page 257
5 Multivariate U-Statistics......Page 263
5.1.1 One Sample Multivariate U-Statistics......Page 264
5.1.2 General K Sample Multivariate U-Statistics......Page 287
5.2 Models for Longitudinal Study Designs......Page 293
5.2.1 Inference in the Absence of Missing Data......Page 294
5.2.2 Inference Under MCAR......Page 298
5.2.3 Inference Under MAR......Page 308
5.3 Exercises......Page 318
6 Functional Response Models......Page 323
6.1 Limitations of Linear Response Models......Page 325
6.2 Models with Functional Responses......Page 330
6.2.1 Models for Group Comparisons......Page 331
6.2.2 Models for Regression Analysis......Page 340
6.3 Model Estimation......Page 347
6.3.1 Inference for Models for Group Comparison......Page 349
6.3.2 Inference for Models for Regression Analysis......Page 361
6.4 Inference for Longitudinal Data......Page 366
6.4.1 Inference Under MCAR......Page 367
6.4.2 Inference Under MAR......Page 374
6.5 Exercises......Page 381
References......Page 387
Subject Index......Page 389
