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B0608
Title: Detection and identification in functional settings Authors:  Antonio Cuevas - Autonomous University of Madrid (Spain) [presenting]
Ricardo Fraiman - Universidad de la Republica (Uruguay)
Abstract: A model is considered in which $n$ random functions $X_i$ are independently observed in such a way that a block of them (whose set of indices $C_n$ is supposed to belong to a given class ${\mathcal C}_n$ of subsets of $\{1,\ldots,n\}$) might have a distribution $G$ different from the (known) distribution $F$ of the remaining observations. In this setting, we consider both, the detection problem of testing $H_0: F=G$ versus $H_1: F\neq G$ and the identification problem of estimating (when $H_1$ is accepted) the set $C_n$ of exceptional indices where the distribution of the $X_i$ is different from that of the bulk'' of the data. These problems of detection and identification of sparse segments arises, for example, in genomic studies. Most literature on the topic is focused on the one-dimensional case, often assuming in addition that the $X_i$ are Gaussian. We tackle the problem for the case of functional data. We show that the Radon-Nikodym (RN) derivatives for the distributions of stochastic processes (especially in the Gaussian case) are a useful tool for these purposes. In short, our proposal is to take the data'' to the real line, using a suitable RN-derivative, and then to use a detection-identification procedure based on the Kolmogorov-Smirnov (or the Cramer-von Mises) statistic.