39 research outputs found
Uniform in bandwidth exact rates for a class of kernel estimators
Given an i.i.d sample , taking values in \RRR^{d'}\times \RRR^d,
we consider a collection Nadarya-Watson kernel estimators of the conditional
expectations \EEE(+d_g(z)\mid Z=z), where belongs to a
compact set H\subset \RRR^d, a Borel function on \RRR^{d'} and
are continuous functions on \RRR^d. Given two
bandwidth sequences h_n<\wth_n fulfilling mild conditions, we obtain an exact
and explicit almost sure limit bounds for the deviations of these estimators
around their expectations, uniformly in g\in\GG,\;z\in H and h_n\le h\le
\wth_n under mild conditions on the density , the class \GG, the kernel
and the functions . We apply this result to prove
that smoothed empirical likelihood can be used to build confidence intervals
for conditional probabilities \PPP(Y\in C\mid Z=z), that hold uniformly in
z\in H,\; C\in \CC,\; h\in [h_n,\wth_n]. Here \CC is a Vapnik-Chervonenkis
class of sets.Comment: Published in the Annals of the Institute of Statistical Mathematics
Volume 63, p. 1077-1102 (2011
An Invariance Principle of G-Brownian Motion for the Law of the Iterated Logarithm under G-expectation
The classical law of the iterated logarithm (LIL for short)as fundamental
limit theorems in probability theory play an important role in the development
of probability theory and its applications. Strassen (1964) extended LIL to
large classes of functional random variables, it is well known as the
invariance principle for LIL which provide an extremely powerful tool in
probability and statistical inference. But recently many phenomena show that
the linearity of probability is a limit for applications, for example in
finance, statistics. As while a nonlinear expectation--- G-expectation has
attracted extensive attentions of mathematicians and economists, more and more
people began to study the nature of the G-expectation space. A natural question
is: Can the classical invariance principle for LIL be generalized under
G-expectation space? This paper gives a positive answer. We present the
invariance principle of G-Brownian motion for the law of the iterated logarithm
under G-expectation