Goodness of fit tests for functional data
∗J.A. Cuesta-Albertos
Departamento de Matem´aticas, Estad´ıstica y Computaci´on, Universidad de Cantabria, Spain
In [?] some results containing sufficient conditions in order that a probability distri- bution to be determined for some one-dimensional marginals were provided; the results being valid even in the case the distribution is defined on an infinite dimensional Hilbert space IH. In particular, there it was shown the following.
Given P and Q two Borel probability distributions on IH, let us denote IE(P, Q) := {h∈IH :Ph =Qh},
where Ph denotes the projection of P on the linear subspace generated by the vector h.
Let µ be a continuous distribution on IH. If µ[IE(P, Q)] > 0 and P satisfies some regularity condition, then P and Q coincide.
Regularity condition in this result consists of, basically, inP being determined by their moments.
Interest of this result relies in the fact that it allows to construct consistent goodness of fit tests for functional distributions based on the analysis of a randomly chosen one- dimensional projections.
In this talk we will analyze this result in order to show how to apply it to construct goodness of fit tests.
References
[1] CUESTA-ALBERTOS, J.A., FRAIMAN, R. and RANSFORD, T. (2004). A sharp form of the Cramr-Wold theorem. Preprint
∗Research partially supported by the Spanish Ministerio de Ciencia y Tecnolog´ıa, grant BFM2002-04430-C02-02.
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