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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.