Preface
Contents
1 The ฮต-ฮด Arguments
1.1 Introduction
1.2 Getting used to the ฮต-ฮด 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 integration
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
2.6 Case study: ฯ2-test
2.7 Summary and additional results
2.8 Exercises
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 CornishโFisher 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 PrasadโRao method
4.9 Exercises
5 Inequalities
5.1 Introduction
5.2 Numerical inequalities
5.2.1 The convex function inequality
5.2.2 Hรถlder'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 Case study: A variance inequality
5.8 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 GlivenkoโCantelli 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 FayโHerriot 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 MetropolisโHastings algorithm
15.4 Monte Carlo EM algorithm
15.5 Convergence rates of Gibbs samplers
15.6 Exercises
16 Random Matrix Theory
16.1 Introduction
16.2 Fundamental theorems of RMT
16.3 Large covariance matrices
16.4 High-dimensional linear models
16.5 Genome-wide association study
16.6 Application to time series
16.7 Exercises
Appendix A
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