Testing for no effect by cosine series methods (Q2711688)

The nonparametric regression model \(y_i=\phi_0+f(t_i)+\varepsilon_i\) with fixed effects is considered, where \(t_i=(2i-1)/(2n)\), \(i=1,\dots,n\), \(\varepsilon_i\sim N(0,1)\), \(f\) is an unknown function, \(\int_0^1 f(t)dt=0\). The hypothesis \(H_0: f\equiv 0\) is tested against general alternatives using sample (cosine) Fourier coefficients \(\hat \phi_j =\sqrt{2} n^{-1}\sum_{i=1}^n\cos(j\pi t_i)y_i \). Different statistics based on \(\hat \phi_j\) are considered, e.g., \(B_n=\sum_{j=1}^{n-1}n\hat\phi_j^2/(j\pi)^2\). The power of the obtained tests is compared with the ``directional'' test based on the statistics \(V_n=\left(\sum_{i=1}^n f( t_i)y_i\right)^2(\sum_{i=1}^n f(t_i)^2)^{-1}\) which provides the most powerful test if \(f\) is known. NEWLINENEWLINENEWLINETo compare the performance of the tests the author uses the intermediate asymptotic efficiency (ARE) approach. Let \(\varphi_V(n,\alpha,f)\) be a power of an \(\alpha\)-level test based on the statistic \(V\) computed by a sample of size \(n\) on the alternative \(f\). Then NEWLINE\[NEWLINEn_{V_1V_2}(n,f,\alpha)=\inf \{m:\varphi_{V_2}(m+k,\alpha,f)>\varphi_{V_1}(n,\alpha,f)\;\forall k=1,2\dots\},NEWLINE\]NEWLINE and the efficiency of \(V_2\) relative to \(V_1\) is \(e_{V_1V_2}=n_{V_1V_2}(n,f,\alpha)/n\). This efficiency is investigated as \(n\to\infty\) for local alternatives \(f_n=n^{-a/ 2}(f(t)-n^{-1}\sum_{j=1}^nf(t_j))\) and levels \(\alpha_n\to 0\) with \(\log\alpha_n=o(n^\tau)\). The \(\tau\)-intermediate ARE is NEWLINE\[NEWLINEE_{V_1V_2}(f)=\underset{n\to\infty} {\text{lim}}e_{V_1V_2}(n,f_n,\alpha_n),\;\text{e.g.,} E_{BV}=(\sum_{j=1}^\infty \phi_j^2/j^2)(\sum_{j=1}^\infty \phi_j^2)^{-1}.NEWLINE\]NEWLINE Results of simulations are presented.