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Table of Contents
Intro
Preface
Acknowledgments
Contents
List of Acronyms
List of Nomenclatures
List of Figures
List of Tables
I. Introduction to Big Data
1. Examples of Big Data
1.1. Multivariate Data
1.2. Categorical Data
1.3. Environmental Data
1.4. Genetic Data
1.5. Time Series Data
1.6. Ranking Data
1.7. Social Network Data
1.8. Symbolic Data
1.9. Image Data
II. Statistical Inference for Big Data
2. Basic Concepts in Probability
2.1. Pearson System of Distributions
2.2. Modes of Convergence
2.3. Multivariate Central Limit Theorem
2.4. Markov Chains
3. Basic Concepts in Statistics
3.1. Parametric Estimation
3.2. Hypothesis Testing
3.3. Classical Bayesian Statistics
4. Multivariate Methods
4.1. Matrix Algebra
4.2. Multivariate Analysis as a Generalization of Univariate Analysis
4.2.1. The General Linear Model
4.2.2. One Sample Problem
4.2.3. Two-Sample Problem
4.3. Structure in Multivariate Data Analysis
4.3.1. Principal Component Analysis
4.3.2. Factor Analysis
4.3.3. Canonical Correlation
4.3.4. Linear Discriminant Analysis
4.3.5. Multidimensional Scaling
4.3.6. Copula Methods
5. Nonparametric Statistics
5.1. Goodness-of-Fit Tests
5.2. Linear Rank Statistics
5.3. U Statistics
5.4. Hoeffding's Combinatorial Central Limit Theorem
5.5. Nonparametric Tests
5.5.1. One-Sample Tests of Location
5.5.2. Confidence Interval for the Median
5.5.3. Wilcoxon Signed Rank Test
5.6. Multi-Sample Tests
5.6.1. Two-Sample Tests for Location
5.6.2. Multi-Sample Test for Location
5.6.3. Tests for Dispersion
5.7. Compatibility
5.8. Tests for Ordered Alternatives
5.9. A Unified Theory of Hypothesis Testing
5.9.1. Umbrella Alternatives
5.9.2. Tests for Trend in Proportions
5.10. Randomized Block Designs
5.11. Density Estimation
5.11.1. Univariate Kernel Density Estimation
5.11.2. The Rank Transform
5.11.3. Multivariate Kernel Density Estimation
5.12. Spatial Data Analysis
5.12.1. Spatial Prediction
5.12.2. Point Poisson Kriging of Areal Data
5.13. Efficiency
5.13.1. Pitman Efficiency
5.13.2. Application of Le Cam's Lemmas
5.14. Permutation Methods
6. Exponential Tilting and Its Applications
6.1. Neyman Smooth Tests
6.2. Smooth Models for Discrete Distributions
6.3. Rejection Sampling
6.4. Tweedie's Formula: Univariate Case
6.5. Tweedie's Formula: Multivariate Case
6.6. The Saddlepoint Approximation and Notions of Information
7. Counting Data Analysis
7.1. Inference for Generalized Linear Models
7.2. Inference for Contingency Tables
7.3. Two-Way Ordered Classifications
7.4. Survival Analysis
7.4.1. Kaplan-Meier Estimator
7.4.2. Modeling Survival Data
8. Time Series Methods
8.1. Classical Methods of Analysis
8.2. State Space Modeling
9. Estimating Equations
Preface
Acknowledgments
Contents
List of Acronyms
List of Nomenclatures
List of Figures
List of Tables
I. Introduction to Big Data
1. Examples of Big Data
1.1. Multivariate Data
1.2. Categorical Data
1.3. Environmental Data
1.4. Genetic Data
1.5. Time Series Data
1.6. Ranking Data
1.7. Social Network Data
1.8. Symbolic Data
1.9. Image Data
II. Statistical Inference for Big Data
2. Basic Concepts in Probability
2.1. Pearson System of Distributions
2.2. Modes of Convergence
2.3. Multivariate Central Limit Theorem
2.4. Markov Chains
3. Basic Concepts in Statistics
3.1. Parametric Estimation
3.2. Hypothesis Testing
3.3. Classical Bayesian Statistics
4. Multivariate Methods
4.1. Matrix Algebra
4.2. Multivariate Analysis as a Generalization of Univariate Analysis
4.2.1. The General Linear Model
4.2.2. One Sample Problem
4.2.3. Two-Sample Problem
4.3. Structure in Multivariate Data Analysis
4.3.1. Principal Component Analysis
4.3.2. Factor Analysis
4.3.3. Canonical Correlation
4.3.4. Linear Discriminant Analysis
4.3.5. Multidimensional Scaling
4.3.6. Copula Methods
5. Nonparametric Statistics
5.1. Goodness-of-Fit Tests
5.2. Linear Rank Statistics
5.3. U Statistics
5.4. Hoeffding's Combinatorial Central Limit Theorem
5.5. Nonparametric Tests
5.5.1. One-Sample Tests of Location
5.5.2. Confidence Interval for the Median
5.5.3. Wilcoxon Signed Rank Test
5.6. Multi-Sample Tests
5.6.1. Two-Sample Tests for Location
5.6.2. Multi-Sample Test for Location
5.6.3. Tests for Dispersion
5.7. Compatibility
5.8. Tests for Ordered Alternatives
5.9. A Unified Theory of Hypothesis Testing
5.9.1. Umbrella Alternatives
5.9.2. Tests for Trend in Proportions
5.10. Randomized Block Designs
5.11. Density Estimation
5.11.1. Univariate Kernel Density Estimation
5.11.2. The Rank Transform
5.11.3. Multivariate Kernel Density Estimation
5.12. Spatial Data Analysis
5.12.1. Spatial Prediction
5.12.2. Point Poisson Kriging of Areal Data
5.13. Efficiency
5.13.1. Pitman Efficiency
5.13.2. Application of Le Cam's Lemmas
5.14. Permutation Methods
6. Exponential Tilting and Its Applications
6.1. Neyman Smooth Tests
6.2. Smooth Models for Discrete Distributions
6.3. Rejection Sampling
6.4. Tweedie's Formula: Univariate Case
6.5. Tweedie's Formula: Multivariate Case
6.6. The Saddlepoint Approximation and Notions of Information
7. Counting Data Analysis
7.1. Inference for Generalized Linear Models
7.2. Inference for Contingency Tables
7.3. Two-Way Ordered Classifications
7.4. Survival Analysis
7.4.1. Kaplan-Meier Estimator
7.4.2. Modeling Survival Data
8. Time Series Methods
8.1. Classical Methods of Analysis
8.2. State Space Modeling
9. Estimating Equations