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 regression and the right-censoring model with monotone hazard rate. The criterion for testing is an L p-...

The traditional maximum likelihood unimodal density estimator (Grenander (1956)) pieces together two isotonic density estimators at a known mode. It is discontinuous at the mode, and does not directly adapt to the case of unknown mode. This paper presents an alternative unimodal density estimator in the form of a generalized isotonic regression on a partial order which is continuous at the mode...

We investigate the limit behavior of the L k-distance between a decreasing density f and its nonparametric maximum likelihood es-timatorˆfn for k ≥ 1. Due to the inconsistency ofˆfn at zero, the case k = 2.5 turns out to be a kind of transition point. We extend asymp-totic normality of the L1-distance to the L k-distance for 1 ≤ k < 2.5, and obtain the analogous limiting result for a modificati...

In this paper, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are ARMA processes. Thereby, the sampling distance is fixed and driving Lévy process has at least a finite fourth moment. Under different assumptions on function moments of derive central limit theorem periodogram. Either assumpt...

We study in this paper some nonparametric inference problems where the nonparametric maximum likelihood estimator (NPMLE) are not well defined. However, if we enlarge the parameter space, the NPMLE will be well defined. We propose to gradually shrink the enlarged parameter space by placing more and more restrictions on the parameter space, producing a sequence of (envelope) estimators. The appr...

The term "Pattern Theory" was introduced by Ulf Grenander in the 70s as a name for a field of applied mathematics which gave a theoretical setting for a large number of related ideas, techniques and results from fields such as computer vision, speech recognition, image and acoustic signal processing, pattern recognition and its statistical side, neural nets and parts of artificial intelligence ...

Vision can be posed as a statistical learning and inference problem. As an over-simplified account, let W be a description of the outside scene in terms of “what is where,” let I be the retina image, and let p(W, I) be the joint distribution ofW and I. Then visual learning is to learn p(W, I) from training data, and visual perception is to infer W from I based on p(W |I). There are two major sc...