In nonparametric and high-dimensional statistical models, the classical Gauss-Fisher-Le Cam theory of the optimality of maximum likelihood and Bayesian posterior inference does not apply, and new foundations and ideas have been developed in the past several decades. This book gives a coherent account of the statistical theory in infinite-dimensional parameter spaces. The mathematical foundations include self-contained 'mini-courses' on the theory of Gaussian and empirical processes, on approximation and wavelet theory and on the basic theory of function spaces. The theory of statistical inference in such models-hypothesis testing, estimation and confidence sets-is then presented within the minimax paradigm of decision theory. This includes the basic theory of convolution kernel and projection estimation, as well as Bayesian nonparametrics and nonparametric maximum likelihood estimation. In the final chapter, the theory of adaptive inference in nonparametric models is developed, including Lepski's method, wavelet thresholding and adaptive confidence regions for self-similar functions. EVARIST GINÉ (1944-2015) was the head of the Department of Mathematics at the University of Connecticut. Giné was a distinguished mathematician who worked on mathematical statistics and probability in infinite dimensions. He was the author of two books and more than a hundred articles.

Let f n denote the usual kernel density estimator in several dimensions. It is shown that if {a n } is a regular band sequence, K is a bounded square integrable kernel of several variables, satisfying some additional mild conditions ((K 1) below), and if the data consist of an i.i.d. sample from a distribution possessing a bounded density f with respect to Lebesgue measure on R d , then lim sup n→∞ na d n log a −1 n sup t ∈R d |f n (t) − Ef n (t)| C f ∞ K 2 (x) dx a.s. for some absolute constant C that depends only on d. With some additional but still weak conditions, it is proved that the above sequence of normalized suprema converges a.s. to 2d f ∞ K 2 (x) dx. Convergence of the moment generating functions is also proved. Neither of these results require f to be strictly positive. These results improve upon, and extend to several dimensions, results by Silverman [13] for univariate densities.

Given a sample from some unknown continuous density $f:\mathbb{R}\to\mathbb{R}$, we construct adaptive confidence bands that are honest for all densities in a "generic" subset of the union of $t$-H\"older balls, $0

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