Based on empirical Levy-type concentration functions a new graphical representation of the ML-density estimator under order restrictions is given. This representation generalizes the well-known representation of the Grenander estimator of a monotone density as the slope of the least concave majorant of the empirical distribution function. From the given representation it follows that a density ...

In various statistical model, such as density estimation and estimation of regression curves or hazard rates, monotonicity constraints can arise naturally. A frequently encountered problem in nonparametric statistics is to estimate a monotone density function f on a compact interval. A known estimator for density function of f under the restriction that f is decreasing, is Grenander estimator, ...

We establish distributional limit theorems for the shape statistics of a concave majorant (i.e. fluctuations its length, supremum, time it is attained and value at T) Lévy process on [0,T] as T→∞. The scale length other statistics, well their asymptotic dependence, vary significantly with tail behaviour measure. key tool in proofs recent representation all processes using stick-breaking represe...

The purpose of this paper is to highlight some hidden Markovian structure the concave majorant Brownian motion. Several distributional identities are implied by joint law a standard one-dimensional motion B and its almost surely unique K on [0,∞). In particular, distribution 2Kt−Bt that R5(t), where R5 5-dimensional Bessel process with R5(0)=0. 2K−B shares number other properties R5, we conject...

We provide a representation of the maximal difference between a standard Brownian bridge and its concave majorant on the unit interval, from which we deduce expressions for the distribution and density functions and moments of this difference. This maximal difference has an application in nonparametric statistics where it arises in testing monotonicity of a density or regression curve.

Kiefer and Wolfowitz [14] showed that if F is a strictly curved concave distribution function (corresponding to a strictly monotone density f), then the Maximum Likelihood Estimator F̂n, which is, in fact, the least concave majorant of the empirical distribution function Fn, differs from the empirical distribution function in the uniform norm by no more than a constant times (n logn)2/3 almost s...

In this paper we investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate n 1 3 . The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown nonincreasing density function f on [0,∞), is a prototypical example. We focus on this example and explore different approaches to cons...

In this paper, we study the least concave majorant of a stochastic process, and more precisely the set of times where its slope varies, which we call the extremal superior times. We establish the negligibility of this set, namely the extremal superior set, for certain classes of processes, including Lévy processes, their integrated processes, and Itô processes. We examine more closely the case ...

The familiar bijections between the representations of permutations as words and as products of cycles have a natural class of “data driven” extensions that permit us to use purely combinatorial means to obtain precise probabilistic information about the geometry of random walks. In particular, we show that the algorithmic bijection of Bohnenblust and Spitzer can be used to obtain means, varian...

the effective factors of forward speed, cylinder rotation, and clearance between a combine’s cylinder and concave on wheat kernel breakage and seed germination were studied in karaj area using a 955 model john dear combine. this study was conducted on three levels of adjustment for each of the variables, 1.2, 1.8 and 2.5 km/hr for forward speed, 800, 900, 1000 rpm for cylinder rotation as well ...