Bi-*-Concave Distributions.

We introduce new shape-constrained classes of distribution functions on , the bi-*-concave classes. In parallel to results of Dümbgen et al. (2017) for what they called the class of bi-log-concave distribution functions, we show that every -concave density has a bi-*-concave distribution function for * ≤ /( + 1).

The Bennett-Orlicz Norm.

van de Geer and Lederer ( 157(1-2), 225-250, 2013) introduced a new Orlicz norm, the Bernstein-Orlicz norm, which is connected to Bernstein type inequalities. Here we introduce another Orlicz norm, the Bennett-Orlicz norm, which is connected to Bennett type inequalities. The new Bennett-Orlicz norm yields inequalities for expectations of maxima which are potentially somewhat tighter than those resulting from the Bernstein-Orlicz norm when they are both applicable.

Entropy of Convex Functions on ℝ .

Let Ω be a bounded closed convex set in ℝ with non-empty interior, and let 𝒞 (Ω) be the class of convex functions on Ω with -norm bounded by 1. We obtain sharp estimates of the -entropy of 𝒞 (Ω) under (Ω) metrics, 1 ≤ < ≤ ∞. In particular, the results imply that the universal lower bound is also an upper bound for all -polytopes, and the universal upper bound of [Formula: see text] for [Formula: see text] is attained by the closed unit ball.

Exponential bounds for the hypergeometric distribution.

We establish exponential bounds for the hypergeometric distribution which include a finite sampling correction factor, but are otherwise analogous to bounds for the binomial distribution due to León and Perron ( (2003) 345-354) and Talagrand ( (1994) 28-76). We also extend a convex ordering of Kemperman's ( (1973) 149-164) for sampling without replacement from populations of real numbers between zero and one: a population of all zeros or ones (and hence yielding a hypergeometric distribution in the upper bound) gives the extreme case.

Finite sampling inequalities: an application to two-sample Kolmogorov-Smirnov statistics.

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 complements recent results by Wei and Dudley (2012) 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 et al.

A law of the iterated logarithm for Grenander's estimator.

In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If () > 0, () 0, and is continuous in a neighborhood of , then [Formula: see text]almost surely where [Formula: see text]here [Formula: see text] is the two-sided Strassen limit set on [Formula: see text]. The proof relies on laws of the iterated logarithm for local empirical processes, Groeneboom's switching relation, and properties of Strassen's limit set analogous to distributional properties of Brownian motion.

APPROXIMATION AND ESTIMATION OF -CONCAVE DENSITIES VIA RÉNYI DIVERGENCES.

In this paper, we study the approximation and estimation of -concave densities via Rényi divergence. We first show that the approximation of a probability measure by an -concave density exists and is unique via the procedure of minimizing a divergence functional proposed by [ (2010) 2998-3027] if and only if admits full-dimensional support and a first moment. We also show continuity of the divergence functional in : if → in the Wasserstein metric, then the projected densities converge in weighted metrics and uniformly on closed subsets of the continuity set of the limit.

GLOBAL RATES OF CONVERGENCE OF THE MLES OF LOG-CONCAVE AND -CONCAVE DENSITIES.

We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and -concave densities on ℝ. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than when -1 < < ∞ where = 0 corresponds to the log-concave case. We also show that the MLE does not exist for the classes of -concave densities with < -1.

On convex least squares estimation when the truth is linear.

We prove that the convex least squares estimator (LSE) attains a pointwise rate of convergence in any region where the truth is linear. In addition, the asymptotic distribution can be characterized by a modified invelope process. Analogous results hold when one uses the derivative of the convex LSE to perform derivative estimation.

Z-estimation and stratified samples: application to survival models.

The infinite dimensional Z-estimation theorem offers a systematic approach to joint estimation of both Euclidean and non-Euclidean parameters in probability models for data. It is easily adapted for stratified sampling designs. This is important in applications to censored survival data because the inverse probability weights that modify the standard estimating equations often depend on the entire follow-up history.

Information bounds for Gaussian copulas.

Often of primary interest in the analysis of multivariate data are the copula parameters describing the dependence among the variables, rather than the univariate marginal distributions. Since the ranks of a multivariate dataset are invariant to changes in the univariate marginal distributions, rank-based estimators are natural candidates for semiparametric copula estimation. Asymptotic information bounds for such estimators can be obtained from an asymptotic analysis of the rank likelihood, i.

Chernoff's density is log-concave.

We show that the density of = argmax{ - }, sometimes known as Chernoff's density, is log-concave. We conjecture that Chernoff's density is strongly log-concave or "super-Gaussian", and provide evidence in support of the conjecture.

WEIGHTED LIKELIHOOD ESTIMATION UNDER TWO-PHASE SAMPLING.

We develop asymptotic theory for weighted likelihood estimators (WLE) under two-phase stratified sampling without replacement. We also consider several variants of WLEs involving estimated weights and calibration. A set of empirical process tools are developed including a Glivenko-Cantelli theorem, a theorem for rates of convergence of -estimators, and a Donsker theorem for the inverse probability weighted empirical processes under two-phase sampling and sampling without replacement at the second phase.

A general semiparametric Z-estimation approach for case-cohort studies.

Case-cohort design, an outcome-dependent sampling design for censored survival data, is increasingly used in biomedical research. The development of asymptotic theory for a case-cohort design in the current literature primarily relies on counting process stochastic integrals. Such an approach, however, is rather limited and lacks theoretical justification for outcome-dependent weighted methods due to non-predictability.

Log-Concavity and Strong Log-Concavity: a review.

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on ℝ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013).

Global rates of convergence of the MLE for multivariate interval censoring.

We establish global rates of convergence of the Maximum Likelihood Estimator (MLE) of a multivariate distribution function on ℝ in the case of (one type of) "interval censored" data. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than (log ) for γ = (5 - 4)/6.

Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas.

Do there exist circular and spherical copulas in [Formula: see text]? That is, do there exist circularly symmetric distributions on the unit disk in [Formula: see text] and spherically symmetric distributions on the unit ball in [Formula: see text], d ≥ 3, whose one-dimensional marginal distributions are uniform? The answer is yes for d = 2 and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for d ≥ 4. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in [Formula: see text] by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in [Formula: see text] are also described, and determined explicitly for d = 2.

Nonparametric estimation of multivariate scale mixtures of uniform densities.

Suppose that U = (U(1), … , U(d)) has a Uniform ([0, 1](d)) distribution, that Y = (Y(1), … , Y(d)) has the distribution G on [Formula: see text], and let X = (X(1), … , X(d)) = (U(1)Y(1), … , U(d)Y(d)). The resulting class of distributions of X (as G varies over all distributions on [Formula: see text]) is called the Scale Mixture of Uniforms class of distributions, and the corresponding class of densities on [Formula: see text] is denoted by [Formula: see text]. We study maximum likelihood estimation in the family [Formula: see text].

NONPARAMETRIC ESTIMATION OF MULTIVARIATE CONVEX-TRANSFORMED DENSITIES.

We study estimation of multivariate densities p of the form p(x) = h(g(x)) for x ∈ ℝ(d) and for a fixed monotone function h and an unknown convex function g. The canonical example is h(y) = e(-y) for y ∈ ℝ; in this case, the resulting class of densities [Formula: see text]is well known as the class of log-concave densities. Other functions h allow for classes of densities with heavier tails than the log-concave class.

A local maximal inequality under uniform entropy.

We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a small constant δ. The bound is expressed in the uniform entropy integral of the class at δ. The bound yields a rate of convergence of minimum contrast estimators when applied to the modulus of continuity of the contrast functions.

Estimation of a k-monotone density: characterizations, consistency and minimax lower bounds.

The classes of monotone or convex (and necessarily monotone) densities on ℝ(+) can be viewed as special cases of the classes of k-monotone densities on ℝ(+). These classes bridge the gap between the classes of monotone (1-monotone) and convex decreasing (2-monotone) densities for which asymptotic results are known, and the class of completely monotone (∞-monotone) densities on ℝ(+). In this paper we consider non-parametric maximum likelihood and least squares estimators of a k-monotone density g(0).

Nonparametric estimation of a convex bathtub-shaped hazard function.

In this paper, we study the nonparametric maximum likelihood estimator (MLE) of a convex hazard function. We show that the MLE is consistent and converges at a local rate of n(2/5) at points x(0) where the true hazard function is positive and strictly convex. Moreover, we establish the pointwise asymptotic distribution theory of our estimator under these same assumptions.

Computation of nonparametric convex hazard estimators via profile methods.

This paper proposes a profile likelihood algorithm to compute the nonparametric maximum likelihood estimator of a convex hazard function. The maximisation is performed in two steps: First the support reduction algorithm is used to maximise the likelihood over all hazard functions with a given point of minimum (or antimode). Then it is shown that the profile (or partially maximised) likelihood is quasi-concave as a function of the antimode, so that a bisection algorithm can be applied to find the maximum of the profile likelihood, and hence also the global maximum.