Features
Describes the major statistical techniques for inferring model parameters, with a focus on the MLE and QMLE
Introduces concepts of nonparametric statistics, including smoothing splines
Covers HMM models, including Gaussian linear, switching Markovian, and nonlinear state space models
Present direct likelihood inference techniques and the EM algorithm
Uses R for numerical examples and provides a dedicated R package
Solutions manual available upon qualifying course adoption
This text emphasizes nonlinear models for a course in time series analysis. After introducing stochastic processes, Markov chains, Poisson processes, and ARMA models, the authors cover functional autoregressive, ARCH, threshold AR, and discrete time series models as well as several complementary approaches. They discuss the main limit theorems for Markov chains, useful inequalities, statistical techniques to infer model parameters, and GLMs. Moving on to HMM models, the book examines filtering and smoothing, parametric and nonparametric inference, advanced particle filtering, and numerical methods for inference.
Author(s): Randal Douc, Eric Moulines, David Stoffer
Series: Chapman & Hall/CRC Texts in Statistical Science
Edition: 1
Publisher: Chapman and Hall/CRC
Year: 2014
Language: English
Commentary: Better Covers Selected, Repeated Covers Removed, Better Bookmarked & Paginated
Pages: xx+531
Tags: Библиотека;Компьютерная литература;R;
Preface
Frequently Used Notation
Part I Foundations
Chapter 1 Linear Models
Stochastic processes
The covariance world
Second-order stationary processes
Spectral representation
Wold decomposition
Linear processes
What are linear Gaussian processes?
ARMA models
Prediction
Estimation
The multivariate cases
Time domain
Frequency domain
Numerical examples
Exercises
Chapter 2 Linear Gaussian State Space Models
Model basics
Filtering, smoothing, and forecasting
Maximum likelihood estimation
Newton–Raphson
EM algorithm
Smoothing splines and the Kalman smoother
Asymptotic distribution of the MLE
Missing data modifications
Structural component models
State-space models with correlated errors
ARMAX models
Regression with autocorrelated errors
Exercises
Chapter 3 Beyond Linear Models
Nonlinear non-Gaussian data
Volterra series expansion
Cumulants and higher-order spectra
Bilinear models
Conditionally heteroscedastic models
Threshold ARMA models
Functional autoregressive models
Linear processes with infinite variance
Models for counts
Integer valued models
Generalized linear models
Numerical examples
Exercises
Chapter 4 Stochastic Recurrence Equations
The Scalar Case
Strict stationarity
Weak stationarity
GARCH(1, 1)
The Vector Case
Strict stationarity
Weak stationarity
GARCH(p, q)
Iterated random function
Strict stationarity
Weak stationarity
Exercises
Part II Markovian Models
Chapter 5 Markov Models: Construction and Definitions
Markov chains: Past, future, and forgetfulness
Kernels
Homogeneous Markov chain
Canonical representation
Invariant measures
Observation-driven models
Iterated random functions
MCMC methods
Metropolis-Hastings algorithm
Gibbs sampling
Exercises
Chapter 6 Stability and Convergence
Uniform ergodicity
Total variation distance
Dobrushin coefficient
The Doeblin condition
Examples
V-geometric ergodicity
V-total variation distance
V-Dobrushin coefficient
Drift and minorization conditions
Examples
Some proofs
Endnotes
Exercises
Chapter 7 Sample Paths and Limit Theorems
Law of large numbers
Dynamical system and ergodicity
Markov chain ergodicity
Central limit theorem
Deviation inequalities for additive functionals
Rosenthal type inequality
Concentration inequality
Some proofs
Exercises
Chapter 8 Inference for Markovian Models
Likelihood inference
Consistency and asymptotic normality of the MLE
Consistency
Asymptotic normality
Observation-driven models
Bayesian inference
Some proofs
Endnotes
Exercises
Part III State Space and Hidden Markov Models
Chapter 9 Non-Gaussian and Nonlinear State Space Models
Definitions and basic properties
Discrete-valued state space HMM
Continuous-valued state-space models
Conditionally Gaussian linear state-space models
Switching processes with Markov regimes
Filtering and smoothing
Discrete-valued state-space HMM
Continuous-valued state-space HMM
Endnotes
Exercises
Chapter 10 Particle Filtering
Importance sampling
Sequential importance sampling
Sampling importance resampling
Algorithm description
Resampling techniques
Particle filter
Sequential importance sampling
Auxiliary sampling
Convergence of the particle filter
Exponential deviation inequalities
Time-uniform bounds
Endnotes
Exercises
Chapter 11 Particle Smoothing
Poor man's smoother algorithm
FFBSm algorithm
FFBSi algorithm
Smoothing functionals
Particle independent Metropolis-Hastings
Particle Gibbs
Convergence of the FFBSm and FFBSi algorithms
Exponential deviation inequality
Asymptotic normality
Time uniform bounds
Endnotes
Exercises
Chapter 12 Inference for Nonlinear State Space Models
Monte Carlo maximum likelihood estimation
Particle approximation of the likelihood function
Particle stochastic gradient
Particle Monte Carlo EM algorithms
Particle stochastic approximation EM
Bayesian analysis
Gaussian linear state space models
Gibbs sampling for NLSS model
Particle marginal Markov chain Monte Carlo
Particle Gibbs algorithm
Endnotes
Exercises
Chapter 13 Asymptotics of the MLE for NLSS
Strong consistency of the MLE
Forgetting the initial distribution
Approximation by conditional likelihood
Strong consistency
Identifiability of mixture densities
Asymptotic normality
Convergence of the observed information
Limit distribution of the MLE
Endnotes
Exercises
Part IV Appendices
Appendix Some Mathematical Background
Some measure theory
Some probability theory
Appendix Martingales
Definitions and elementary properties
Limits theorems
Appendix Stochastic Approximation
Robbins–Monro algorithm: Elementary results
Stochastic gradient
Stepsize selection and averaging
The Kiefer–Wolfowitz procedure
Appendix Data Augmentation
The EM algorithm in the incomplete data model
The Fisher and Louis identities
Monte Carlo EM algorithm
Stochastic approximation EM
Convergence of the EM algorithm
Convergence of the MCEM algorithm
Convergence of perturbed dynamical systems
Convergence of the MCEM algorithm
References
Index
