Scale space consistency of piecewise constant least squares estimators -- another look at the regressogram

We study the asymptotic behavior of piecewise constant least squares regression estimates, when the number of partitions of the estimate is penalized. We show that the estimator is consistent in the relevant metric if the signal is in $L^2([0,1])$, the space of c\`{a}dl\`{a}g functions equipped with the Skorokhod metric or $C([0,1])$ equipped with the supremum metric. Moreover, we consider the family of estimates under a varying smoothing parameter, also called scale space. We prove convergence of the empirical scale space towards its deterministic target.Comment: Published at http://dx.doi.org/10.1214/074921707000000274 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Affine equivariant rank-weighted L-estimation of multivariate location

In the multivariate one-sample location model, we propose a class of flexible robust, affine-equivariant L-estimators of location, for distributions invoking affine-invariance of Mahalanobis distances of individual observations. An involved iteration process for their computation is numerically illustrated.Comment: 16 pages, 4 figures, 6 table

Estimation of location and covariance with high breakdown point

Electrical Engineering, Mathematics and Computer Scienc

The limit process of the difference between the empirical distribution function and its concave majorant

We consider the process –Fn, being the difference between the empirical distribution function Fn and its least concave majorant , corresponding to a sample from a decreasing density. We extend Wang's result on pointwise convergence of –Fn and prove that this difference converges as a process in distribution to the corresponding process for two-sided Brownian motion with parabolic drift

Distribution of Global Measures of Deviation Between the Empirical Distribution Function and Its Concave Majorant

We investigate the distribution of some global measures of deviation between the empirical distribution function and its least concave majorant. In the case that the underlying distribution has a strictly decreasing density, we prove asymptotic normality for several L k -type distances. In the case of a uniform distribution, we also establish their limit distribution together with that of the supremum distance. It turns out that in the uniform case, the measures of deviation are of greater order and their limit distributions are different.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc

A central limit theorem for the Hellinger loss of Grenander-type estimators

We consider Grenander-type estimators for a monotone function (Formula presented.), obtained as the slope of a concave (convex) estimate of the primitive of λ. Our main result is a central limit theorem for the Hellinger loss, which applies to estimation of a probability density, a regression function or a failure rate. In the case of density estimation, the limiting variance of the Hellinger loss turns out to be independent of λ.Statistic

Central limit theorems for the Lp-error of smooth isotonic estimators

We investigate the asymptotic behavior of the Lp-distance betweena monotone function on a compact interval and a smooth estimatorof this function. Our main result is a central limit theorem for the Lp-errorof smooth isotonic estimators obtained by smoothing a Grenander-typeestimator or isotonizing the ordinary kernel estimator. As a preliminary resultwe establish a similar result for ordinary kernel estimators. Our resultsare obtained in a general setting, which includes estimation of a monotonedensity, regression function and hazard rate. We also perform a simulationstudy for testing monotonicity on the basis of the L2-distance between thekernel estimator and the smoothed Grenander-type estimator.Statistic