Band 22, Heft 1 - 1

We consider the median regression model X k = θ( x k ) + ξ k , where the unknown signal θ : [0,1] → ℝ, is assumed to belong to a Hölder smoothness class, the ξ k s are independent, but not necessarily identically distributed, noises with zero median. The distribution of the noise is assumed to be unknown and satisfying some weak conditions. Possible noise distributions may have heavy tails, so that, for example, the expectation of noises does not exist. This implies that in general linear methods (for example, kernel method) cannot be applied directly in this situation. On the basis of a preliminary recursive estimator, we construct certain variables Y k s, called pseudovalues which do not depend on the noise distribution, and derive an asymptotic expansion (uniform over a certain class of noise distributions): Y k = θ ( x k ) + ∊ k + r k , where ∊ k s are binary random variables and the remainder terms r k s are negligible. This expansion mimics the nonparametric regression model with binary noises. In so doing, we reduce our original observation model with “bad” (heavy-tailed) noises effectively to the nonparametric regression model with binary noises.

There are many asymptotically first order efficient estimators in the problem of estimating the invariant density of an ergodic diffusion process nonparametrically. To distinguish between them, we consider the problem of asymptotically second order minimax estimation of this density based on a sample path observation up to the time T . It means that we have two problems. The first one is to find a lower bound on the second order risk of any estimator. The second one is to construct an estimator, which attains this lower bound. We carry out this program (bound + estimator) following Pinsker’s approach. If the parameter set is a subset of the Sobolev ball of smoothness k  > 1 and radius R  > 0, the second order minimax risk is shown to behave as − T −2 k /(2 k −1) Π̂( k , R ) for large values of T . The constant Π̂( k , R ) is given explicitly.

For rounding arbitrary probabilities on finitely many categories to rational proportions, the multiplier method with standard rounding stands out. Sainte-Laguë showed in 1910 that the method minimizes a goodness-of-fit criterion that nowadays classifies as a chi-square divergence. Assuming the given probabilities to be uniformly distributed, we derive the limiting law of the Sainte-Laguë divergence, first when the rounding accuracy increases, and then when the number of categories grows large. The latter limit turns out to be a Lévy-stable distribution.

In this paper we construct estimators for linear functionals of bivariate distributions with parametric marginals. Our construction generalizes the construction of efficient estimators given by Peng and Schick (2002) for the case of known marginals. More precisely, we show that in their construction the fixed and known orthonormal bases for the Hilbert spaces of square integrable functions with zero means can now be replaced by estimated orthonormal bases using a n 1/2 -consistent estimator of the parameter.