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Title: Family-wise error rate on domain subsets: A unified framework for local inference in functional data analysis Authors:  Konrad Abramowicz - Umea University (Sweden) [presenting]
Alessia Pini - Università Cattolica del Sacro Cuore (Italy)
Lina Schelin - Umea University (Sweden)
Simone Vantini - Politecnico di Milano (Italy)
Sara Sjostedt-de Luna - Umea University (Sweden)
Abstract: A functional test of the hypotheses H0 against H1 (e.g., a test on parameters of a functional-on-scalar linear model) is considered where the aim is to select the parts of the domain where H0 is violated, while controlling the probability of false discoveries. It is straightforward to define an unadjusted p-value function, associating a p-value with each point of the domain. Such a function only point-wise controls the probability of type I error, so it cannot be used for domain-selection purposes, since it would not provide any global control of the probability of type-I error. That is why the definition of an adjusted p-value function provided with a stronger control is often required. We require the control of the probability of falsely rejecting the null hypothesis on subsets of the domain (control of the family-wise error rate, FWER on subsets). We compare different methods to define the adjusted p-value function. The methods that we discuss belong to a general set of methods based on the following steps: a family $S$ of subsets of the domain is defined; the restriction of the null hypothesis is tested on every element of the family; the adjusted p-value of each point is computed as the maximum p-value of the tests of every element containing that point. We consider several methods where the choice of $S$ is either fixed or data-driven.