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.