A new nonparametric procedure for testing monotonicity of a regression mean is proposed. The test is shown to have prescribed asymptotic level and good asymptotic power. It is based on the supremum distance from an empirical process to its least concave majorant and is very easily implementable....

In this article, we develop a test for the null hypothesis that a real-valued function belongs to a given parametric set against the non-parametric alternative that it is monotone, say decreasing. The method is described in a general model that covers the monotone density model, the monotone...

The least squares estimator of a discrete distribution under the constraint of convexity is introduced. Its existence and uniqueness are shown and consistency and rate of convergence are established. Moreover it is shown that it always outperforms the classical empirical estimator in terms of...

Probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of "maximizing the sharpness of the predictive...

We prove a second Marshall inequality for adaptive convex density estimation via least squares. The result completes the first inequality proved recently by Dümbgen et al. [2007. Marshall's lemma for convex density estimation. IMS Lecture Notes--Monograph Series, submitted for publication....

The classes of monotone or convex (and necessarily monotone) densities on inline image can be viewed as special cases of the classes of k-monotone densities on inline image. These classes bridge the gap between the classes of monotone (1-monotone) and convex decreasing (2-monotone) densities for...

We consider the problem of mixing k random variables where each of the k components results from shifting a common random variable X0 with a certain probability. We show that if X0 admits a density that is a Pólya frequency function with E[X0]=0, then k, a1,…,ak and π1,…,πk are...