Reconstruction of the real 3D shape of the SARS-CoV-2 virus.

The photographs of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus taken by electron transmission microscopy and cryoelectron microscopy provide only a 2D silhouette. The viruses appear to look like distorted circles. The present paper questions the real shape of the SARS-CoV-2 virus and makes an attempt to give an answer.

Age-stratified discrete compartment model of the COVID-19 epidemic with application to Switzerland.

Compartmental models enable the analysis and prediction of an epidemic including the number of infected, hospitalized and deceased individuals in a population. They allow for computational case studies on non-pharmaceutical interventions thereby providing an important basis for policy makers. While research is ongoing on the transmission dynamics of the SARS-CoV-2 coronavirus, it is important to come up with epidemic models that can describe the main stages of the progression of the associated COVID-19 respiratory disease.

Letter to the editor comments on Groparu-Cojocaru and Doray (2013).

Although estimating the five parameters of an unknown Generalized Normal Laplace (GNL) density by minimizing the distance between the empirical and true characteristic functions seems appealing, the approach cannot be advocated in practice. This conclusion is based on extensive numerical simulations in which a fast minimization procedure delivers deceiving estimators with values that are quite far away from the truth. These findings can be predicted by the very large values obtained for the true asymptotic variances of the estimators of the five parameters of the true GNL density.

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.

ON THE GRENANDER ESTIMATOR AT ZERO.

We establish limit theory for the Grenander estimator of a monotone density near zero. In particular we consider the situation when the true density f(0) is unbounded at zero, with different rates of growth to infinity. In the course of our study we develop new switching relations using tools from convex analysis.

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).

Limit Distribution Theory for Maximum Likelihood Estimation of a Log-Concave Density.

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, i.e. a density of the form f(0) = exp varphi(0) where varphi(0) is a concave function on R.