Abstract
Exploiting an expansion for analytic functions of operators, the asymptotic distribution of an estimator of the functional regression parameter is obtained in a rather simple way; the result is applied to testing linear hypotheses. The expansion is also used to obtain a quick proof for the asymptotic optimality of a functional classification rule, given Gaussian populations.
Highlights
Certain functions of the covariance operator such as the square root of a regularized inverse are important components of many statistics employed for functional data analysis
Two further applications of the approximation in 1.1 will be given, both related to functional regression
Hall and Horowitz 1 have shown that the IMSE of their estimator, based on a Tikhonov type regularized inverse, is rate optimal
Summary
Certain functions of the covariance operator such as the square root of a regularized inverse are important components of many statistics employed for functional data analysis. When the perturbation Σ − Σ in the present case commutes with Σ the expansion 1.1 can already be found in Dunford & Schwartz 10, Chapter VII , and the derivative does reduce to the numerical derivative This condition is fulfilled only in very special cases, for instance, when the random function, whose covariance operator is Σ, is a second order stationary process on the unit interval. In this situation, the eigenfunctions are known and only the eigenvalues are to be estimated. Functional time series are considered in Bosq ; see Mas
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