Locally adaptive fitting of semiparametric models to nonstationary time series. (Q1879516)

A sequence of stochastic processes \(\{X_ {t,T}\}_ {t=1,\dots , T}\) is called locally stationary with transfer function \(A^ {0}\) and (smooth) trend \(\mu \) if \[ X_ {t,T}=\mu (t/T)+ \int _ {-\pi }^ {\pi } A_ {t,T}^ {0}(\omega ) \exp \{i\omega t\}\, d\xi (\omega ) \] under the conditions that (i) \(\xi (\omega )\) is a process with orthonormal increments and (ii) there exist a smooth function \(A(u,\omega )\) and a constant \(K\) such that \[ \sup _ {t, \omega } | A_ {t,T}^ {0}(\omega ) - A (t/T, \omega)| \leq KT^ {-1}. \] The function \(f(u,\omega )=| A(u, \omega )| ^ {2}\) is called the time-varying spectral density of \(\{X_ {t,T}\}\). It is assumed that \(f(u, \omega )\) depends on a \(p\)-dimensional parameter function \(\theta (u)\), \(u\in [0,1]\). The mean function is estimated by a usual kernel estimator whereas locally adaptive estimation by wavelet thresholding is applied to the components of \(\theta \). The fundamental tool in this procedure is the preperiodogram \(J_ T(t/T, \lambda)\), which uses the single product \(X_ {[t+0.5-k/2],T} X_ {[t+0.5+k/2],T}\) as a kind of local estimator of the covariance of lag \(k\) at time \(t\). The preperiodogram is connected with the ordinary periodogram \(I_ T (\lambda )\) by the formula \(I_ T (\lambda )= T^{-1} \sum _ {t=1}^ T J_ T(t/T,\lambda)\). It is proved that the proposed estimators converge to the corresponding target quantities with a rate that matches the optimal rate in some smoothness classes of the parameters.