Large Sample Techniques for Statistics.

Author
Jiang, Jiming [Browse]
Format
Book
Language
English
Εdition
2nd ed.
Published/​Created
  • Cham : Springer International Publishing AG, 2022.
  • ©2022.
Description
1 online resource (689 pages)

Details

Series
Springer Texts in Statistics Ser. [More in this series]
Source of description
Description based on publisher supplied metadata and other sources.
Contents
  • Intro -- Preface -- Contents -- 1 The ε-δ Arguments -- 1.1 Introduction -- 1.2 Getting used to the ε-δ 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: χ2-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 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 Hölder'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 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 -- 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.
ISBN
9783030916954 ((electronic bk.))
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