The FEXP estimator for potentially non-stationary linear time series. (Q1766049)

Let \(B\) be the backshift operator and \(X=(X_t)\) a process such that \(Y=(1-B)^p X\) is stationary for some integer \(p\). Assume that \(Y\) has a spectral density \(f(x)=| 1-\text{e}^{ix}| ^{-2d} f^*(x)\) where \(f^*\) is an even, non-negative, continuous function with \(f^*(0)\neq 0\) and \(-p-1/2<d<1/2\). This means that there exists a stationary process \(Y^*=(I-B)^{p+d} X\) with spectral density \(f^*\). Semiparametric fractional exponential (FEXP) estimators of \(d\) are considered. They are based on FEXP models of order \(q\) that have the spectral density \(f\) with \(f^*(x)=\exp\{\sum _{j=0}^{q-1} \theta _j h_j(x)\}\) where \(h_0(x)=1/\sqrt {2\pi }\) and \(h_j(x)=\cos (jx)\left /\sqrt {\pi }\right .\) for \(j\geq 1\). A method of pooling the tapered periodogram leads to more efficient estimators of \(d\). Under general assumptions asymptotic normality of the estimators is established. Finally, minimax rate-optimality and feasible nearly rate optimal estimators in the Gaussian case are considered.