Spectral Estimation with Random Truncation

Abstract

Let $f(\omega), -\pi \leqq \omega \leqq \pi$ be the spectral density function of a discrete coordinate real-valued time series, stationary to order four. Assume that the covariance function $r(k)$ is such that $-\log r^2(k) \sim Ck^\gamma$, and $-\log(r^2(k + 1)/r^2(k)) \sim C_\gamma k^{\gamma-1}$, as $k \rightarrow \infty$, for some $C, \gamma, 0 < C, \gamma < \infty$. Then there exists a non-random sequence $t(n)$ which is such that the estimator $f^\ast(\omega) \equiv (2\pi)^{-1} \sum^{t(n)}_{k=-t(n)} (1 - n^{-1}|k|)\hat{r}(k) e^{ik\omega}$ is efficient where $\hat{r}(k) \equiv (n - |k|)^{-1} \sum^{n-|k|}_{j=1}(X(j) - \overline{X})(X(j + |k|) - \overline{X}), \overline{X} = n^{-1} \sum^n_{j=1} X(j)$, and an estimator $\hat{f}(\omega)$ is said to be efficient if $\lim_{n\rightarrow\infty}2\pi E\int^\pi_{-\pi} (\hat{f}(\omega) - f(\omega))^2 d\omega/I^2_{\min}(n) = 1$, where $I^2_{\min}(n)$ is the smallest integrated mean squared error which can be achieved using an estimator of the form $\tilde{f}(\omega) = (2\pi)^{-1} \sum^{n-1}_{k=-(n-1)} a(k,n)\hat{r}(k)e^{ik\omega}$, where $a(k,n)$ is nonrandom. In general a sequence $t(n)$ which is efficient for one covariance function is inefficient for another. A class of estimators $\hat{f}(\omega)$ is presented which are of the form $\hat{f}(\omega) = (2\pi)^{-1} \sum^{\hat{t}(n)}_{k=-\hat{t}(n)}(1 - n^{-1}|k|)\hat{r}(k)e^{ik\omega}$, where $\hat{t}(n)$ is a function of the observations. In an appropriate sense $\hat{t}(n)$ "estimates" $t(n)$. For any covariance function satisfying the above conditions $\sup_{-\pi\leqq\omega\leqq\pi}|\hat{f}(\omega) - \hat{f}^\ast(\omega)|/I_{\min}(n) \rightarrow 0$, in probability, where $f^\ast(\omega)$ is the unattainable efficient truncation estimator.