9,500 research outputs found
A uniform functional law of the logarithm for the local empirical process
We prove a uniform functional law of the logarithm for the local empirical
process. To accomplish this we combine techniques from classical and abstract
empirical process theory, Gaussian distributional approximation and probability
on Banach spaces. The body of techniques we develop should prove useful to the
study of the strong consistency of d-variate kernel-type nonparametric function
estimators.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000024
Uniform in bandwidth consistency of kernel-type function estimators
We introduce a general method to prove uniform in bandwidth consistency of
kernel-type function estimators. Examples include the kernel density estimator,
the Nadaraya-Watson regression estimator and the conditional empirical process.
Our results may be useful to establish uniform consistency of data-driven
bandwidth kernel-type function estimators.Comment: Published at http://dx.doi.org/10.1214/009053605000000129 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Randomly Weighted Self-normalized L\'evy Processes
Let be a bivariate L\'evy process, where is a subordinator
and is a L\'evy process formed by randomly weighting each jump of
by an independent random variable having cdf . We investigate the
asymptotic distribution of the self-normalized L\'evy process at 0
and at . We show that all subsequential limits of this ratio at 0
() are continuous for any nondegenerate with finite expectation if
and only if belongs to the centered Feller class at 0 (). We also
characterize when has a non-degenerate limit distribution at 0 and
.Comment: 32 page
A note on a maximal Bernstein inequality
We show somewhat unexpectedly that whenever a general Bernstein-type maximal
inequality holds for partial sums of a sequence of random variables, a maximal
form of the inequality is also valid.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ304 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
The limit distribution of ratios of jumps and sums of jumps of subordinators
Let be a driftless subordinator, and let denote its jump sequence on interval . Put for the -trimmed subordinator.
In this note we characterize under what conditions the limiting distribution of
the ratios and
exist, as or .Comment: 14 page
Couplings and Strong Approximations to Time Dependent Empirical Processes Based on I.I.D. Fractional Brownian Motions
We define a time dependent empirical process based on i.i.d.~fractional
Brownian motions and establish Gaussian couplings and strong approximations to
it by Gaussian processes. They lead to functional laws of the iterated
logarithm for this process.Comment: To appear in the Journal of Theoretical Probability. 37 pages.
Corrected version. The results on quantile processes are taken out and it
will appear elsewher
On the Breiman conjecture
Let be positive, nondegenerate, i.i.d. random
variables, and independently let be i.i.d. random
variables. In this note we show that whenever
converges in distribution to nondegenerate limit for some ,
in a specified class of distributions , then necessarily
belongs to the domain of attraction of a stable law with index less than 1. The
class contains those nondegenerate with a finite second
moment and those in the domain of attraction of a stable law with index
Asymptotic normality of plug-in level set estimates
We establish the asymptotic normality of the -measure of the symmetric
difference between the level set and a plug-in-type estimator of it formed by
replacing the density in the definition of the level set by a kernel density
estimator. Our proof will highlight the efficacy of Poissonization methods in
the treatment of large sample theory problems of this kind.Comment: Published in at http://dx.doi.org/10.1214/08-AAP569 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Revisiting two strong approximation results of Dudley and Philipp
We demonstrate the strength of a coupling derived from a Gaussian
approximation of Zaitsev (1987a) by revisiting two strong approximation results
for the empirical process of Dudley and Philipp (1983), and using the coupling
to derive extended and refined versions of them.Comment: Published at http://dx.doi.org/10.1214/074921706000000824 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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