Jiming Jiang. Large Sample Techniques for Statistics

Springer, 2010. — 609 p. — (Springer Texts in Statistics). — ISBN 978-1-4419-6826-5.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 ε-δ ArgumentsGetting used to the ε-δ arguments
More examples
Case study: Consistency of MLE in the i.i.d. case
Some useful results
Exercises
Modes of ConvergenceConvergence in probability
Almost sure convergence
Convergence in distribution
L^p convergence and related topics
Case study: χ²-test
Summary and additional results
Exercises
Big O, Small o, and the Unspecified cBig O and small o for sequences and functions
Big O and small o for vectors and matrices
Big O and small o for random quantities
The unspecified c and other similar methods
Case study: The baseball problem.
Case study: Likelihood ratio for a clustering problem
Exercises
Asymptotic ExpansionsTaylor expansion
Edgeworth expansion; method of formal derivation
Other related expansions
Some elementary expansions
Laplace approximation
Case study: Asymptotic distribution of the MLE
Case study: The Prasad–Rao method
Exercises
InequalitiesNumerical inequalities
Matrix inequalities
Integral/moment inequalities
Probability inequalities
Case study: Some problems on existence of moments
Exercises
Sums of Independent Random VariablesThe weak law of large numbers
The strong law of large numbers
The central limit theorem
The law of the iterated logarithm
Further results
Case study: The least squares estimators
Exercises
Empirical ProcessesGlivenko–Cantelli theorem and statistical functionals
Weak convergence of empirical processes
LIL and strong approximation
Bounds and large deviations
Non-i.i.d. observations
Empirical processes indexed by functions
Case study: Estimation of ROC curve and ODC
Exercises
MartingalesExamples and simple properties
Two important theorems of martingales
Martingale laws of large numbers
A martingale central limit theorem and related topic
Convergence rate in SLLN and LIL
Invariance principles for martingales
Case study: CLTs for quadratic forms
Case study: Martingale approximation
Exercises
Time and Spatial SeriesAutocovariances and autocorrelations
The information criteria
ARMA model identification
Strong limit theorems for i.i.d. spatial series
Two-parameter martingale differences
Sample ACV and ACR for spatial series
Case study: Spatial AR models
Exercises
Stochastic ProcessesMarkov chains
Poisson processes
Renewal theory
Brownian motion
Stochastic integrals and diffusions
Case study: GARCH models and financial SDE
Exercises
Nonparametric StatisticsSome classical nonparametric tests
Asymptotic relative efficiency
Goodness-of-fit tests
U-statistics
Density estimation
Exercises
Mixed Effects ModelsREML: Restricted maximum likelihood
Linear mixed model diagnostics
Inference about GLMM
Mixed model selection
Exercises
Small-Area EstimationEmpirical best prediction with binary data
The Fay–Herriot model
Nonparametric small-area estimation
Model selection for small-area estimation
Exercises
Jackknife and BootstrapThe jackknife
Jackknifing the MSPE of EBP
The bootstrap
Bootstrapping time series
Bootstrapping mixed models
Exercises
Markov-Chain Monte CarloThe Gibbs sampler
The Metropolis–Hastings algorithm
Monte Carlo EM algorithm
Convergence rates of Gibbs samplers
Exercises
Appendix
Matrix algebra
Measure and probability
Some results in statistics
List of notation and abbreviations