Large Sample Techniques for Statistics (Springer Texts in Statistics)

✍ Scribed by Jiming Jiang

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 611
• Category: Library

✦ Synopsis

This book offers a comprehensive guide to large sample techniques in statistics. More importantly, it focuses on thinking skills rather than just what formulae to use; it provides motivations, and intuition, rather than detailed proofs; it begins with very simple techniques, and connects theory and applications in entertaining ways. The first five chapters review some of the basic techniques, such as the fundamental epsilon-delta arguments, Taylor expansion, different types of convergence, and inequalities. The next five chapters discuss limit theorems in specific situations of observational data. Each of the first 10 chapters contains at least one section of case study. The last five chapters are devoted to special areas of applications. The sections of case studies and chapters of applications fully demonstrate how to use methods developed from large sample theory in various, less-than-textbook situations. The book is supplemented by a large number of exercises, giving the readers plenty of opportunities to practice what they have learned. The book is mostly self-contained with the appendices providing some backgrounds for matrix algebra and mathematical statistics. The book is intended for a wide audience, ranging from senior undergraduate students to researchers with Ph.D. degrees. A first course in mathematical statistics and a course in calculus are prerequisites.

✦ Table of Contents

Preface
Contents
1 The -d Arguments
1.1 Introduction
1.2 Getting used to the arguments
1.3 More examples
1.4 Case study: Consistency of MLE in the i.i.d. case
1.5 Some useful results
1.5.1 Infinite sequence
1.5.2 Infinite series
1.5.3 Topology
1.5.4 Continuity, differentiation, and intergration
1.6 Exercises
2 Modes of Convergence
2.1 Introduction
2.2 Convergence in probability
2.3 Almost sure convergence
2.4 Convergence in distribution
2.5 Lp convergence and related topics
3 Big O, Small o, and the Unspecified c
3.1 Introduction
3.2 Big O and small o for sequences and functions
3.3 Big O and small o for vectors and matrices
3.4 Big O and small o for random quantities
3.5 The unspecified c and other similar methods
3.6 Case study: The baseball problem
3.7 Case study: Likelihood ratio for a clustering problem
3.8 Exercises
4 Asymptotic Expansions
4.1 Introduction
4.2 Taylor expansion
4.3 Edgeworth expansion; method of formal derivation
4.4 Other related expansions
4.4.1 Fourier series expansion
4.4.2 Cornish–Fisher expansion
4.4.3 Two time series expansions
4.5 Some elementary expansions
4.6 Laplace approximation
4.7 Case study: Asymptotic distribution of the MLE
4.8 Case study: The Prasad–Rao method
4.9 Exercises
5 Inequalities
5.1 Introduction
5.2 Numerical inequalities
5.2.1 The convex function inequality
5.2.2 H¨older’s and related inequalities
5.2.3 Monotone functions and related inequalities
5.3 Matrix inequalities
5.3.1 Nonnegative definite matrices
5.3.2 Characteristics of matrices
5.4 Integral/moment inequalities
5.5 Probability inequalities
5.6 Case study: Some problems on existence of moments
5.7 Exercises
6 Sums of Independent Random Variables
6.1 Introduction
6.2 The weak law of large numbers
6.3 The strong law of large numbers
6.4 The central limit theorem
6.5 The law of the iterated logarithm
6.6 Further results
6.6.1 Invariance principles in CLT and LIL
6.6.2 Large deviations
6.7 Case study: The least squares estimators
6.8 Exercises
7 Empirical Processes
7.1 Introduction
7.2 Glivenko–Cantelli theorem and statistical functionals
7.3 Weak convergence of empirical processes
7.4 LIL and strong approximation
7.5 Bounds and large deviations
7.6 Non-i.i.d. observations
7.7 Empirical processes indexed by functions
7.8 Case study: Estimation of ROC curve and ODC
7.9 Exercises
8 Martingales
8.1 Introduction
8.2 Examples and simple properties
8.3 Two important theorems of martingales
8.3.1 The optional stopping theorem
8.3.2 The martingale convergence theorem
8.4 Martingale laws of large numbers
8.4.1 A weak law of large numbers
8.4.2 Some strong laws of large numbers
8.5 A martingale central limit theorem and related topic
8.6 Convergence rate in SLLN and LIL
8.7 Invariance principles for martingales
8.8 Case study: CLTs for quadratic forms
8.9 Case study: Martingale approximation
8.10 Exercises
9 Time and Spatial Series
9.1 Introduction
9.2 Autocovariances and autocorrelations
9.3 The information criteria
9.4 ARMA model identification
9.5 Strong limit theorems for i.i.d. spatial series
9.6 Two-parameter martingale differences
9.7 Sample ACV and ACR for spatial series
9.8 Case study: Spatial AR models
9.9 Exercises
10 Stochastic Processes
10.1 Introduction
10.2 Markov chains
10.3 Poisson processes
10.4 Renewal theory
10.5 Brownian motion
10.6 Stochastic integrals and diffusions
10.7 Case study: GARCH models and financial SDE
10.8 Exercises
11 Nonparametric Statistics
11.1 Introduction
11.2 Some classical nonparametric tests
11.3 Asymptotic relative efficiency
11.4 Goodness-of-fit tests
11.5 U-statistics
11.6 Density estimation
11.7 Exercises
12 Mixed Effects Models
12.1 Introduction
12.2 REML: Restricted maximum likelihood
12.3 Linear mixed model diagnostics
12.4 Inference about GLMM
12.5 Mixed model selection
12.6 Exercises
13 Small-Area Estimation
13.1 Introduction
13.2 Empirical best prediction with binary data
13.3 The Fay–Herriot model
13.4 Nonparametric small-area estimation
13.5 Model selection for small-area estimation
13.6 Exercises
14 Jackknife and Bootstrap
14.1 Introduction
14.2 The jackknife
14.3 Jackknifing the MSPE of EBP
14.4 The bootstrap
14.5 Bootstrapping time series
14.6 Bootstrapping mixed models
14.7 Exercises
15 Markov-Chain Monte Carlo
15.1 Introduction
15.2 The Gibbs sampler
15.3 The Metropolis–Hastings algorithm
15.4 Monte Carlo EM algorithm
15.5 Convergence rates of Gibbs samplers
15.6 Exercises
A Appendix
A.1 Matrix algebra
A.1.1 Numbers associated with a matrix
A.1.2 Inverse of a matrix
A.1.3 Kronecker products
A.1.4 Matrix differentiation
A.1.5 Projection
A.1.6 Decompositions of matrices and eigenvalues
A.2 Measure and probability
A.2.1 Measures
A.2.2 Measurable functions
A.2.3 Integration
A.2.4 Distributions and random variables
A.2.5 Conditional expectations
A.2.6 Conditional distributions
A.3 Some results in statistics
A.3.1 The multivariate normal distribution
A.3.2 Maximum likelihood
A.3.3 Exponential family and generalized linear models
A.3.4 Bayesian inference
A.3.5 Stationary processes
A.4 List of notation and abbreviations
References
Index