OPTIMALITY OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN FIRST‐ORDER AUTOREGRESSIVE PROCESSES

Piotr W. Mikulski and Michael J. Monsour

Journal of Time Series Analysis, 1991, vol. 12, issue 3, 237-253

Abstract: Abstract. Let ρρ* be the maximum likelihood estimator (MLE) of the parameter ρ in the first‐order autoregressive process with normal errors. The problem of optimality in the sense of weighted squared error is considered rather than moments of asymptotic distributions. Many unbiased estimators can be constructed, but the two‐dimensional sufficient statistic is incomplete. It is shown that dEρ* ‐ ρ→ 0 uniformly in ρ, and that dE(ρ*)/dρ→ 1 for all |ρ| > 1. For |ρ| > 1, it is known that the asymptotic distribution of {I(ρ)}12(ρ* ‐ ρ), where I(ρ) is the Fisher information, is Cauchy. It follows that the Cramèr‐Rao inequality will not yield useful results for investigating the limit of exact efficiency of any asymptotically unbiased estimator, including the MLE. For all |ρ|

Date: 1991
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