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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
fda (R)
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