
Theory and Methods of Statistics
Resources
Description
Key Features
- Codifies foundational information in many core areas of statistics into a comprehensive and definitive resource
- Serves as an excellent text for select master’s and PhD programs, as well as a professional reference
- Integrates numerous examples to illustrate advanced concepts
- Includes many probability inequalities useful for investigating convergence of statistical procedures
Readership
Graduate (Masters/PhD) students and research statisticians.
Table of Contents
1: Probability Theory
- Abstract
- 1.1 Random Experiments and Their Outcomes
- 1.2 Set Theory
- 1.3 Axiomatic Definition of Probability
- 1.4 Some Simple Propositions
- 1.5 Equally Likely Outcomes in Finite Sample Space
- 1.6 Conditional Probability and Independence
- 1.7 Random Variables and Their Distributions
- 1.8 Expected Value, Variance, Covariance, and Correlation Coefficient
- 1.9 Moments and the Moment Generating Function
- 1.10 Independent Random Variables and Conditioning When There Is Dependence
- 1.11 Transforms of Random Variables and Their Distributions
- Exercises
2: Some Common Probability Distributions
- Abstract
- 2.1 Discrete Distributions
- 2.2 Continuous Distributions
- Exercises
3: Infinite Sequences of Random Variables and Their Convergence Properties
- Abstract
- 3.1 Introduction
- 3.2 Modes of Convergence
- 3.3 Probability Inequalities
- 3.4 Asymptotic Normality: The Central Limit Theorem and Its Generalizations
- Exercises
4: Basic Concepts of Statistical Inference
- Abstract
- 4.1 Population and Random Samples
- 4.2 Parametric and Nonparametric Models
- 4.3 Problems of Statistical Inference
- 4.4 Statistical Decision Functions
- 4.5 Sufficient Statistics
- 4.6 Optimal Decision Rules
- Exercises
5: Point Estimation in Parametric Models
- Abstract
- 5.1 Optimality Under Unbiasedness, Squared-Error Loss, UMVUE
- 5.2 Lower Bound for the Variance of an Unbiased Estimator
- 5.3 Equivariance
- 5.4 Bayesian Estimation Using Conjugate Priors
- 5.5 Methods of Estimation
- Exercises
6: Hypothesis Testing
- Abstract
- 6.1 Early History
- 6.2 Basic Concepts
- 6.3 Simple Null Hypothesis vs Simple Alternative: Neyman-Pearson Lemma
- 6.4 UMP Tests for One-Sided Hypotheses Against One-Sided Alternatives in Monotone Likelihood Ratio Families
- 6.5 Unbiased Tests
- 6.6 Generalized Neyman-Pearson Lemma
- 6.7 UMP Unbiased Tests for Two-Sided Problems
- 6.8 Locally Best Tests
- 6.9 UMP Unbiased Tests in the Presence of Nuisance Parameters: Similarity and Completeness
- 6.10 The p-Value: Another Way to Report the Result of a Test
- 6.11 Sequential Probability Ratio Test
- 6.12 Confidence Sets
- Exercises
7: Methods Based on Likelihood and Their Asymptotic properties
- Abstract
- 7.1 Asymptotic Properties of the MLEs: Consistency and Asymptotic Normality
- 7.2 Likelihood Ratio Test
- 7.3 Asymptotic Properties of MLE and LRT Based on Independent Nonidentically Distributed Data
- 7.4 Frequency X2
- Exercises
8: Distribution-Free Tests for Hypothesis Testing in Nonparametric Families
- Abstract
- 8.1 Ranks and Order Statistics
- 8.2 Locally Most Powerful Rank Tests
- 8.3 Tests Based on Empirical Distribution Function
- Exercises
9: Curve Estimation
- Abstract
- 9.1 Introduction
- 9.2 Density Estimation
- 9.3 Regression Estimation
- 9.4 Nearest Neighbor Approach
- 9.5 Curve Estimation in Higher Dimension
- 9.6 Curve Estimation Using Local Polynomials
- 9.7 Estimation of Survival Function and Hazard Rates Under Random Right-Censoring
- Exercises
10: Statistical Functionals and Their Use in Robust Estimation
- Abstract
- 10.1 Introduction
- 10.2 Functional Delta Method
- 10.3 The L-Estimators
- 10.4 The M-Estimators
- 10.5 A Relation Between L-Estimators and M-Estimators
- 10.6 The Remainder Term Rn
- 10.7 The Jackknife and the Bootstrap
- Exercises
11: Linear Models
- Abstract
- 11.1 Introduction
- 11.2 Examples of Gauss-Markov Models
- 11.3 Gauss-Markov Models: Estimation
- 11.4 Decomposition of Total Sum of Squares
- 11.5 Estimation Under Linear Restrictions on β
- 11.6 Gauss-Markov Models: Inference
- 11.7 Analysis of Covariance
- 11.8 Model Selection
- 11.9 Some Alternate Methods for Regression
- 11.10 Random- and Mixed-Effects Models
- 11.11 Inference: Examples From Mixed Models
- Exercises
12: Multivariate Analysis
- Abstract
- 12.1 Introduction
- 12.2 Wishart Distribution
- 12.3 The Role of Multivariate Normal Distribution
- 12.4 One-Sample Inference
- 12.5 Two-Sample Problem
- 12.6 One-Factor MANOVA
- 12.7 Two-Factor MANOVA
- 12.8 Multivariate Linear Model
- 12.9 Principal Components Analysis
- 12.10 Factor Analysis
- 12.11 Classification and Discrimination
- 12.12 Canonical Correlation Analysis
- Exercises
13: Time Series
- Abstract
- 13.1 Introduction
- 13.2 Concept of Stationarity
- 13.3 Estimation of the Mean and the Autocorrelation Function
- 13.4 Partial Autocorrelation Function (PACF)
- 13.5 Causality and Invertibility
- 13.6 Forecasting
- 13.7 ARIMA Models and Forecasting
- 13.8 Parameter Estimation
- 13.9 Selection of an Appropriate ARMA model
- 13.10 Spectral Analysis
- Exercises
Appendix A: Results From Analysis and Probability
- A.1 Some Important Results in Integration Theory
- A.2 Convex Functions
- A.3 Stieltjes Integral
- A.4 Characteristic Function, Weak Law of Large Number, and Central Limit Theorem
- A.5 Weak Convergence of Probabilities on C[0,1]
Appendix B: Basic Results From Matrix Algebra
- B.1 Some Elementary Facts
- B.2 Eigenvalues and Eigenvectors
- B.3 Functions of Symmetric Matrices
- B.4 Generalized Eigenvalues
- B.5 Matrix Derivatives
- B.6 Orthogonal Projection
- B.7 Distribution of Quadratic Forms
Product details
- No. of pages: 544
- Language: English
- Copyright: © Academic Press 2016
- Published: May 26, 2016
- Imprint: Academic Press
- eBook ISBN: 9780128041239
- Paperback ISBN: 9780128024409
About the Authors
P.K. Bhattacharya
Affiliations and Expertise
Prabir Burman
Affiliations and Expertise
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