Central Limit Theorems For Local Emprical Processes Near Boundaries of Sets

AMS 2000 subject classifications. 60F05, 60F17, 60G55, 62G30.

Goodness-of-fit problem for errors in nonparametric regression: Distribution free approach

This paper discusses asymptotically distribution free tests for the classical goodness-of-fit hypothesis of an error distribution in nonparametric regression models. These tests are based on the same martingale transform of the residual empirical process as used in the one sample location model. This transformation eliminates extra randomization due to covariates but not due the errors, which is intrinsically present in the estimators of the regression function. Thus, tests based on the transformed process have, generally, better power. The results of this paper are applicable as soon as asymptotic uniform linearity of nonparametric residual empirical process is available. In particular they are applicable under the conditions stipulated in recent papers of Akritas and Van Keilegom and M\"uller, Schick and Wefelmeyer.Comment: Published in at http://dx.doi.org/10.1214/08-AOS680 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Central limit theorems for local empirical processes near boundaries of sets

We define the local empirical process, based on $n$ i.i.d. random vectors in dimension $d$, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that, for these local empirical processes, indexed by classes of sets that vary with $n$ and satisfy certain conditions, an appropriately defined uniform central limit theorem holds. The concept of differentiation of sets in measure is very convenient for developing the results. Some examples and statistical applications are also presented.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ283 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Differentiation of sets in measure

AbstractSuppose F(ε), for each ε∈[0,1], is a bounded Borel subset of Rd and F(ε)→F(0) as ε→0. Let A(ε)=F(ε)▵F(0) be symmetric difference and P be an absolutely continuous measure on Rd. We introduce the notion of derivative of F(ε) with respect to ε, dF(ε)/dε=dA(ε)/dε, such thatddεP(A(ε))|ε=0=Q(ddεA(ε)|ε=0), where Q is another, explicitly described, measure, although not in Rd.We discuss why this sort of derivative is needed to study local point processes in neighbourhood of a set: in short, if sequence of point processes Nn, n=1,2,…, is given on the class of set-valued mappings F={F(⋅)} such that all F(ε) converge to the same F=F(0), then the weak limit of the local processes {Nn(A(ε)),F(ε)∈F} “lives” on the class of derivative sets {dF(ε)/dε|ε=0,F(⋅)∈F}.We compare this notion of the derivative set-valued mapping with other existing notions