Small Sample Effects in Time Series Analysis: A New Asymptotic Theory and a New Estimate

To estimate the parameters of a stationary process, Whittle (1953) introduced an approximation to the Gaussian likelihood function. Although the Whittle estimate is asymptotically efficient, the small sample behavior may be poor if the spectrum of the process contains peaks. We introduce a mathematical model that covers such small sample effects. We prove that the exact maximum likelihood estimate is still optimal in this model, whereas the Whittle estimate and the conditional likelihood estimate are not. Furthermore, we introduce tapered Whittle estimates and prove that these estimates have the same optimality properties as exact maximum likelihood estimates.