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Nonparametric kernel estimators apply to the statistical analysis
of independent or dependent sequences of random variables and for
samples of continuous or discrete processes. The optimization of
these procedures is based on the choice of a bandwidth that
minimizes an estimation error and the weak convergence of the
estimators is proved. This book introduces new mathematical results
on statistical methods for the density and regression functions
presented in the mathematical literature and for functions defining
more complex models such as the models for the intensity of point
processes, for the drift and variance of auto-regressive diffusions
and the single-index regression models.This second edition presents
minimax properties with Lp risks, for a positive real p, and
optimal convergence results for new kernel estimators of function
defining processes: models for multidimensional variables, periodic
intensities, estimators of the distribution functions of censored
and truncated variables, estimation in frailty models, estimators
for time dependent diffusions, for spatial diffusions and for
diffusions with stochastic volatility.
The book is aimed at graduate students and researchers with basic
knowledge of Probability and Integration Theory. It introduces
classical inequalities in vector and functional spaces with
applications to probability. It also develops new extensions of the
analytical inequalities, with sharper bounds and generalizations to
the sum or the supremum of random variables, to martingales and to
transformed Brownian motions. The proofs of many new results are
presented in great detail. Original tools are developed for spatial
point processes and stochastic integration with respect to local
martingales in the plane.This second edition covers properties of
random variables and time continuous local martingales with a
discontinuous predictable compensator, with exponential
inequalities and new inequalities for their maximum variable and
their p-variations. A chapter on stochastic calculus presents the
exponential sub-martingales developed for stationary processes and
their properties. Another chapter devoted itself to the renewal
theory of processes and to semi-Markovian processes, branching
processes and shock processes. The Chapman-Kolmogorov equations for
strong semi-Markovian processes provide equations for their hitting
times in a functional setting which extends the exponential
properties of the Markovian processes.
The book presents advanced methods of integral calculus and
optimization, the classical theory of ordinary and partial
differential equations and systems of dynamical equations. It
provides explicit solutions of linear and nonlinear differential
equations, and implicit solutions with discrete approximations.The
main changes of this second edition are: the addition of
theoretical sections proving the existence and the unicity of the
solutions for linear differential equations on real and complex
spaces and for nonlinear differential equations defined by locally
Lipschitz functions of the derivatives, as well as the
approximations of nonlinear parabolic, elliptic, and hyperbolic
equations with locally differentiable operators which allow to
prove the existence of their solutions; furthermore, the behavior
of the solutions of differential equations under small
perturbations of the initial condition or of the differential
operators is studied.
The book introduces classical inequalities in vector and functional
spaces with applications to probability. It develops new analytical
inequalities, with sharper bounds and generalizations to the sum or
the supremum of random variables, to martingales, to transformed
Brownian motions and diffusions, to Markov and point processes,
renewal, branching and shock processes.In this third edition, the
inequalities for martingales are presented in two chapters for
discrete and time-continuous local martingales with new results for
the bound of the norms of a martingale by the norms of the
predictable processes of its quadratic variations, for the norms of
their supremum and their p-variations. More inequalities are also
covered for the tail probabilities of Gaussian processes and for
spatial processes.This book is well-suited for undergraduate and
graduate students as well as researchers in theoretical and applied
mathematics.
'This is a solid mathematical treatment of some topics in the
analysis of change-point models. The book is intended for graduate
students and scientific researchers using statistics in
practice.'zbMATHThis book provides a detailed exposition of the
specific properties of methods of estimation and test in a wide
range of models with changes. They include parametric and
nonparametric models for samples, series, point processes and
diffusion processes, with changes at the threshold of variables or
at a time or an index of sampling.The book contains many new
results and fills a gap in statistics literature, where the
asymptotic properties of the estimators and test statistics in
singular models are not sufficiently developed. It is suitable for
graduate students and scientific researchers working in the
industry, governmental laboratories and academia.
This book presents advanced methods of integral calculus and the
classical theory of the ordinary and partial differential
equations. It provides explicit solutions of linear and nonlinear
differential equations and implicit solutions with discrete
approximations. Differential equations that could not be explicitly
solved are discussed with special functions such as Bessel
functions. New functions are defined from differential equations.
Laguerre, Hermite and Legendre orthonormal polynomials as well as
several extensions are also considered.It is illustrated by
examples and graphs of functions, with each chapter containing
exercises solved in the last chapter.
An overview of the asymptotic theory of optimal nonparametric tests
is presented in this book. It covers a wide range of topics:
Neyman-Pearson and LeCam's theories of optimal tests, the theories
of empirical processes and kernel estimators with extensions of
their applications to the asymptotic behavior of tests for
distribution functions, densities and curves of the nonparametric
models defining the distributions of point processes and
diffusions. With many new test statistics developed for smooth
curves, the reliance on kernel estimators with bias corrections and
the weak convergence of the estimators are useful to prove the
asymptotic properties of the tests, extending the coverage to
semiparametric models. They include tests built from continuously
observed processes and observations with cumulative intervals.
The book is aimed at graduate students and researchers with basic
knowledge of Probability and Integration Theory. It introduces
classical inequalities in vector and functional spaces with
applications to probability. It also develops new extensions of the
analytical inequalities, with sharper bounds and generalizations to
the sum or the supremum of random variables, to martingales and to
transformed Brownian motions. The proofs of the new results are
presented in great detail.
This book presents a unified approach on nonparametric estimators
for models of independent observations, jump processes and
continuous processes. New estimators are defined and their limiting
behavior is studied. From a practical point of view, the book
expounds on the construction of estimators for functionals of
processes and densities, and provides asymptotic expansions and
optimality properties from smooth estimators. It also presents new
regular estimators for functionals of processes, compares histogram
and kernel estimators, compares several new estimators for
single-index models, and it examines the weak convergence of the
estimators.
The book is intended to undergraduate students, it presents
exercices and problems with rigorous solutions covering the mains
subject of the course with both theory and applications.The
questions are solved using simple mathematical methods: Laplace and
Fourier transforms provide direct proofs of the main convergence
results for sequences of random variables.The book studies a large
range of distribution functions for random variables and processes:
Bernoulli, multinomial, exponential, Gamma, Beta, Dirichlet,
Poisson, Gaussian, Chi2, ordered variables, survival distributions
and processes, Markov chains and processes, Brownian motion and
bridge, diffusions, spatial processes.
The approximation and the estimation of nonparametric functions by
projections on an orthonormal basis of functions are useful in data
analysis. This book presents series estimators defined by
projections on bases of functions, they extend the estimators of
densities to mixture models, deconvolution and inverse problems, to
semi-parametric and nonparametric models for regressions, hazard
functions and diffusions. They are estimated in the Hilbert spaces
with respect to the distribution function of the regressors and
their optimal rates of convergence are proved. Their mean square
errors depend on the size of the basis which is consistently
estimated by cross-validation. Wavelets estimators are defined and
studied in the same models.The choice of the basis, with suitable
parametrizations, and their estimation improve the existing methods
and leads to applications to a wide class of models. The rates of
convergence of the series estimators are the best among all
nonparametric estimators with a great improvement in
multidimensional models. Original methods are developed for the
estimation in deconvolution and inverse problems. The asymptotic
properties of test statistics based on the estimators are also
established.
The book is intended to undergraduate students, it presents
exercices and problems with rigorous solutions covering the mains
subject of the course with both theory and applications.The
questions are solved using simple mathematical methods: Laplace and
Fourier transforms provide direct proofs of the main convergence
results for sequences of random variables.The book studies a large
range of distribution functions for random variables and processes:
Bernoulli, multinomial, exponential, Gamma, Beta, Dirichlet,
Poisson, Gaussian, Chi2, ordered variables, survival distributions
and processes, Markov chains and processes, Brownian motion and
bridge, diffusions, spatial processes.
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