Let $Omega$ be a bounded closed convex set in ${mathbb R}^d$ with non-empty interior, and let ${cal C}_r(Omega)$ be the class of convex functions on $Omega$ with $L^r$-norm bounded by $1$. We obtain sharp estimates of the $epsilon$-entropy of ${cal C
}_r(Omega)$ under $L^p(Omega)$ metrics, $1le p<rle infty$. In particular, the results imply that the universal lower bound $epsilon^{-d/2}$ is also an upper bound for all $d$-polytopes, and the universal upper bound of $epsilon^{-frac{(d-1)}{2}cdot frac{pr}{r-p}}$ for $p>frac{dr}{d+(d-1)r}$ is attained by the closed unit ball. While a general convex body can be approximated by inscribed polytopes, the entropy rate does not carry over to the limiting body. Our results have applications to questions concerning rates of convergence of nonparametric estimators of high-dimensional shape-constrained functions.
We review a finite-sampling exponential bound due to Serfling and discuss related exponential bounds for the hypergeometric distribution. We then discuss how such bounds motivate some new results for two-sample empirical processes. Our development co
mplements recent results by Wei and Dudley (2011) concerning exponential bounds for two-sided Kolmogorov - Smirnov statistics by giving corresponding results for one-sided statistics with emphasis on adjusted inequalities of the type proved originally by Dvoretzky, Kiefer, and Wolfowitz (1956) and by Massart (1990) for one-samp
In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If $f(t_0) > 0$, $f(t_0) < 0$, and $f$ is continuous in a neighborhood of $t_0$, then begin{eqnarray*} limsup_{nrightarrow
infty} left ( frac{n}{2log log n} right )^{1/3} ( widehat{f}_n (t_0 ) - f(t_0) ) = left| f(t_0) f(t_0)/2 right|^{1/3} 2M end{eqnarray*} almost surely where $ M equiv sup_{g in {cal G}} T_g = (3/4)^{1/3}$ and $ T_g equiv mbox{argmax}_u { g(u) - u^2 } $; here ${cal G}$ is the two-sided Strassen limit set on $R$. The proof relies on laws of the iterated logarithm for local empirical processes, Groenebooms switching relation, and properties of Strassens limit set analogous to distributional properties of Brownian motion.
We present a general law of the iterated logarithm for stochastic processes on the open unit interval having subexponential tails in a locally uniform fashion. It applies to standard Brownian bridge but also to suitably standardized empirical distrib
ution functions. This leads to new goodness-of-fit tests and confidence bands which refine the procedures of Berk and Jones (1979) and Owen (1995). Roughly speaking, the high power and accuracy of the latter procedures in the tail regions of distributions are essentially preserved while gaining considerably in the central region.
A bracketing metric entropy bound for the class of Laplace transforms of probability measures on [0,infty) is obtained through its connection with the small deviation probability of a smooth Gaussian process. Our results for the particular smooth Gaussian process seem to be of independent interest.
We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=expvarphi_0$ where $varphi_0$ is a concave function on $mathbb{R}$. The pointwise limiting distributi
ons depend on the second and third derivatives at 0 of $H_k$, the lower invelope of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of $varphi_0=log f_0$ at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode $M(f_0)$ and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.
