Uniform in bandwidth exact rates for a class of kernel estimators

Given an i.i.d sample $(Y_i,Z_i)$, 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 $z$ belongs to a compact set H\subset \RRR^d, $g$ a Borel function on \RRR^{d'} and $c_g(\cdot),d_g(\cdot)$ 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 $f_Z$, the class \GG, the kernel $K$ and the functions $c_g(\cdot),d_g(\cdot)$. 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

Consistency of the kernel density estimator - a survey

Various consistency proofs for the kernel density estimator have been developed over the last few decades. Important milestones are the pointwise consistency and almost sure uniform convergence with a fixed bandwidth on the one hand and the rate of convergence with a fixed or even a variable bandwidth on the other hand. While considering global properties of the empirical distribution functions is sufficient for strong consistency, proofs of exact convergence rates use deeper information about the underlying empirical processes. A unifying character, however, is that earlier and more recent proofs use bounds on the probability that a sum of random variables deviates from its mean

Uniform in Bandwidth Estimation of the Gradient Lines of a Density

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