Odile Pons Knihy

Focusing on nonparametric estimators, the book explores models of independent observations, jump processes, and continuous processes. It introduces new estimators and analyzes their limiting behavior, emphasizing practical applications in constructing estimators for functionals and densities. The text includes asymptotic expansions and optimality properties of smooth estimators, compares histogram and kernel estimators for single-index models, and investigates the weak convergence of the proposed estimators, offering a comprehensive resource for statistical analysis.

Focusing on classical inequalities in vector and functional spaces, this book explores their applications in probability theory. It presents innovative analytical inequalities that provide improved bounds and generalizations for various mathematical concepts, including the sum and supremum of random variables, martingales, and transformed Brownian motions. Additionally, it delves into Markov processes, point processes, and various stochastic processes like renewal, branching, and shock processes, making it a valuable resource for advanced studies in probability and analysis.

Advanced methods of integral calculus and classical theories of ordinary and partial differential equations are explored in this book. It offers explicit solutions for both linear and nonlinear differential equations, alongside implicit solutions using discrete approximations. The text addresses equations that resist explicit solutions, utilizing special functions like Bessel functions. Additionally, it introduces new functions derived from differential equations and examines Laguerre, Hermite, and Legendre orthonormal polynomials, along with various extensions.

Focusing on nonparametric kernel estimators, this book explores their application in statistical analysis for both independent and dependent sequences of random variables. It emphasizes the importance of bandwidth selection to minimize estimation error and demonstrates weak convergence of the estimators. Additionally, it presents new mathematical results related to density and regression functions, as well as more complex models, including point process intensity, auto-regressive diffusion parameters, and single-index regression models.

Advanced methods in integral calculus and optimization are explored, alongside classical theories of ordinary and partial differential equations. The book offers explicit solutions for both linear and nonlinear differential equations, as well as implicit solutions featuring discrete approximations, making it a comprehensive resource for those studying advanced mathematical concepts.

Focusing on advanced statistical methods, this book explores the approximation and estimation of nonparametric functions through projections onto orthonormal function bases. It introduces series estimators that enhance density estimators for various complex models, including mixture, deconvolution, and semi-parametric models. The authors demonstrate optimal convergence rates in Hilbert spaces and analyze mean square errors relative to basis size, utilizing cross-validation for consistent estimation. Additionally, wavelet estimators are examined within these frameworks, providing a comprehensive approach to modern data analysis techniques.

Focusing on the asymptotic theory of optimal nonparametric tests, this book explores Neyman Pearson and LeCam's theories, alongside empirical processes and kernel estimators. It delves into the asymptotic behavior of tests related to distribution functions, densities, and nonparametric models. The text introduces new test statistics for smooth curves, emphasizing kernel estimators with bias corrections and weak convergence. Additionally, it extends to semiparametric models, incorporating tests derived from continuously observed processes and cumulative interval observations.

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.