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Testing hypotheses in the functional linear model. (English) Zbl 1034.62037
An i.i.d. sample \((X_i,Y_i)_{i=1}^n\) is observed, where \(X_i=X_i(t)\) are \(L_2[0,1]\) random elements and \(Y_i=\int_0^1 \psi(t)X_i(t)dt+\varepsilon_i\), \(\varepsilon_i\) are independent of \(X_i\) zero mean r.v.s with \(E\varepsilon_i^2=\sigma^2\) (unknown). The hypotheses \(H_0:\psi=\psi_0\) is tested against \(\psi\not=\psi_0\). \(H_0\) is reduced to the hypothesis \(\psi=0\) which is tested using the empirical cross-covariance operator \(\Delta_n\): for \(x\in L_2[0,1]\), \[ \Delta_n x=n^{-1}\sum_{i=1}^N Y_i\int X_i(t)x(t)\,dt. \] Consistency of the test is demonstrated. Results of simulations are presented.

MSC:
62G10 Nonparametric hypothesis testing
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
Software:
fda (R)
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