308 research outputs found
Local functional principal component analysis
Covariance operators of random functions are crucial tools to study the way
random elements concentrate over their support. The principal component
analysis of a random function X is well-known from a theoretical viewpoint and
extensively used in practical situations. In this work we focus on local
covariance operators. They provide some pieces of information about the
distribution of X around a fixed point of the space x₀. A description of
the asymptotic behaviour of the theoretical and empirical counterparts is
carried out. Asymptotic developments are given under assumptions on the
location of x₀ and on the distributions of projections of the data on the
eigenspaces of the (non-local) covariance operator
Minimax adaptive tests for the Functional Linear model
We introduce two novel procedures to test the nullity of the slope function
in the functional linear model with real output. The test statistics combine
multiple testing ideas and random projections of the input data through
functional Principal Component Analysis. Interestingly, the procedures are
completely data-driven and do not require any prior knowledge on the smoothness
of the slope nor on the smoothness of the covariate functions. The levels and
powers against local alternatives are assessed in a nonasymptotic setting. This
allows us to prove that these procedures are minimax adaptive (up to an
unavoidable \log\log n multiplicative term) to the unknown regularity of the
slope. As a side result, the minimax separation distances of the slope are
derived for a large range of regularity classes. A numerical study illustrates
these theoretical results
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