Results 1 
4 of
4
Representation of gaussian small ball probabilities in l2
, 901
"... Let z = P +∞ i=1 x2i/a 2 i where the xi’s are i.d.d centered with unit variance gaussian random variables and (ai) i∈N an increasing sequence such that P +∞ i=1 a−2 i < +∞. We propose an exponentialintegral representation theorem for the gaussian small ball probability P (z < ε) when ε ↓ 0. W ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Let z = P +∞ i=1 x2i/a 2 i where the xi’s are i.d.d centered with unit variance gaussian random variables and (ai) i∈N an increasing sequence such that P +∞ i=1 a−2 i < +∞. We propose an exponentialintegral representation theorem for the gaussian small ball probability P (z < ε) when ε ↓ 0. We start from a result by MeyerWolf, Zeitouni (1993) and Dembo, MeyerWolf, Zeitouni (1995) who computed this probability by means of series. We prove that P (z < ε) belongs to a class of functions introduced by de Haan, wellknown in extreme value theory, the class Gamma, for which an explicit exponentialintegral representation is available. The converse implication holds under a mild additional assumption. Some applications are underlined in connection with statistical inference for random functions. Keywords: de Haan’s Gamma class, small ball problems, regular variations, gaussian random elements. 1
Local Linear Functional Regression based on Weighted DistanceBased Regression
"... We consider the problem of nonparametrically predicting a scalar response variable y from a functional predictor χ. We have n observations (χi, yi) and we assign a weight wi ∝ K (d(χ, χi)/h) to each χi, where d ( · , · ) is a semimetric, K is a kernel function and h is the bandwidth. Then we fit ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
We consider the problem of nonparametrically predicting a scalar response variable y from a functional predictor χ. We have n observations (χi, yi) and we assign a weight wi ∝ K (d(χ, χi)/h) to each χi, where d ( · , · ) is a semimetric, K is a kernel function and h is the bandwidth. Then we fit a Weighted (Linear) DistanceBased Regression, where the weights are as above and the distances are given by a possibly different semimetric. This approach can be extended to nonparametric predictions from other kind of explanatory variables (e.g., data of mixed type) in a natural way. Key words: Distancebased prediction, functional data analysis, local linear regression, nonparametric regression, weighted regression. 1