Non-parametric curve estimation by wavelet thresholding with locally stationary errors (Q2742762)

The following model is considered: \(X_{t,T} = \mu(t/T) + \varepsilon_{t,T}\), \(t=1,\ldots,T\), where \(\mu(u) \in BV_{[0,1]}\) is a function of bounded variation on \([0, 1]\) such that \(\mu(u)\) belongs to the smoothness Besov class \(B_{p,q}^m\) with \(m, p, q \geq 1.\) The sequence \(\{ \varepsilon_{t,T}\}_{t=1,\ldots,T}\) is locally stationary, that is NEWLINE\[NEWLINE\varepsilon_{t,T} =(2\pi)^{-1/2} \int_{-\pi}^\pi A_{tT}^0 (\omega) \exp \{ i\omega t\} d\xi(\omega), \quad t=1,\ldots,T, NEWLINE\]NEWLINE where \(\xi(\omega)\) is a mean-zero orthonormal increment process on \([-\pi, \pi]\), and there exist a positive constant \(K\) and a regular function \(A(u, \omega)\) such that for all \(T\) NEWLINE\[NEWLINE \sup_{t,\omega} |A_{t,T}^0 (\omega) - A(t/T, \omega)|\leq KT^{-1}. NEWLINE\]NEWLINE The authors consider the following non-linear threshold estimator of the function \(\mu\): NEWLINE\[NEWLINE \hat \mu (u) = \sum_{k=0}^{2j_0 -1} \hat \alpha_k \phi_{j_0 k}(u) + \sum_{j=j_0}^{J-1} \sum_{k=0}^{2^j -1} \tilde\beta_{jk} \psi_{jk}(u), NEWLINE\]NEWLINE where \(\phi_{j_0 k}(u)\), \(\psi_{jk}(u)\) is a wavelet basis, \(J = \log_2(T)\), \(\tilde\beta_{jk} = \delta^{(\cdot)}( \hat\beta_{jk}, \lambda_{jk})\), NEWLINE\[NEWLINE\delta^{(h)}(\hat\beta_{jk}, \lambda_{jk}) = \beta_{jk} I(|\hat\beta_{jk}|\geq \lambda_{jk}),\;\delta^{(s)}(\hat\beta_{jk}, \lambda_{jk})={\text sgn}(\hat\beta_{jk}) (|\hat\beta_{jk}|- \lambda_{jk})_+,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\hat\alpha_k =T^{-1} \sum_t X_{t,T} \phi_{j_0 k}(t/T),\;\hat\beta_{jk} =T^{-1} \sum_t X_{t,T} \psi_{j k}(t/T).NEWLINE\]NEWLINE The authors derive asymptotic formulas for the bias and variance of the empirical wavelet coefficients and establish an \(a^n\) upper bound for the uniform \(L_2\)-risk of the wavelet threshold estimator over a certain smoothness class \(F.\) For example, the authors prove that NEWLINE\[NEWLINE\sup_{\mu \in F}\bigl\{ E\|\hat\mu - \mu\|_{L_2}^2 \bigr\} = O((\log(T)/T)2m/(2m+1)).NEWLINE\]NEWLINE The method is illustrated by a simulated example and by a biostatistical dataset. Measurements of a sheep luteinizing hormone which exhibits a clear non-stationarity in its variance are considered.