We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least (log(n)/n) 1/3 and typically (log(n)/n) 2/5 , whereas the difference between the empirical and estimated distribution function vanishes with rate op(n −1/2 ) under certain regularity assumptions.

We introduce a multiscale test statistic based on local order statistics and spacings that provides simultaneous confidence statements for the existence and location of local increases and decreases of a density or a failure rate. The procedure provides guaranteed finite-sample significance levels, is easy to implement and possesses certain asymptotic optimality and adaptivity properties.Comment: Version 2 is an extended version (Technical report 56, IMSV, Univ. Bern) which is referred to in version 3. Published in at http://dx.doi.org/10.1214/07-AOS521 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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This survey provides a self-contained account of M -estimation of multivariate scatter. In particular, we present new proofs for existence of the underlying M -functionals and discuss their weak continuity and differentiability. This is done in a rather general framework with matrix-valued random variables. By doing so we reveal a connection between Tyler's (1987a) M -functional of scatter and the estimation of proportional covariance matrices. Moreover, this general framework allows us to treat a new class of scatter estimators, based on symmetrizations of arbitrary order. Finally these results are applied to M -estimation of multivariate location and scatter via multivariate t-distributions.MSC 2010 subject classifications: 62G20, 62G35, 62H12, 62H99.

In this paper we show that the family P d (lc) of probability distributions on ℝ d with log-concave densities satisfies a strong continuity condition. In particular, it turns out that weak convergence within this family entails (i) convergence in total variation distance, (ii) convergence of arbitrary moments, and (iii) pointwise convergence of Laplace transforms. In this and several other respects the nonparametric model P d (lc) behaves like a parametric model such as, for instance, the family of all d-variate Gaussian distributions. As a consequence of the continuity result, we prove the existence of nontrivial confidence sets for the moments of an unknown distribution in P d (lc). Our results are based on various new inequalities for log-concave distributions which are of independent interest.

The variance of sO2 measurements between identical devices increased significantly when saturation decreased from the normal level of 97% to the hypoxemic levels of 85% and 75%.

We develop an active set algorithm for the maximum likelihood estimation of a log-concave density based on complete data. Building on this fast algorithm, we indicate an EM algorithm to treat arbitrarily censored or binned data.

Medical telephone counselling is a demanding task requiring competent specialists with dedicated training in communication supported by suitable computer technology. Provided these conditions are in place, computer-assisted telephone triage can be considered to be a safe method of assessing the potential clinical risks of patients' medical conditions.