COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1748 Optimal Estimation under Nonstandard Conditions Werner Ploberger and Peter C.B. Phillips January 2010 We analyze optimality properties of maximum likelihood (ML) and other estimators when
the problem does not necessarily fall within the locally asymptotically normal (LAN)
class, therefore covering cases that are excluded from conventional LAN theory such as
unit root nonstationary time series. The classical Hájek-Le Cam optimality theory is
adapted to cover this situation. We show that the expectation of certain monotone
"bowl-shaped" functions of the squared estimation error are minimized by the ML
estimator in locally asymptotically quadratic situations, which often occur in
nonstationary time series analysis when the LAN property fails. Moreover, we demonstrate a
direct connection between the (Bayesian property of) asymptotic normality of the posterior
and the classical optimality properties of ML estimators. |
