Abstract: It is well known that
the inverse Fisher information is the lower bound for asymptotic uncertainty of
any unbiased consistent estimator. This result is for first order asymptotics.
To study further accuracy of inference, high order behavior of estimators has
received lots of attention, but work on high order information bound is rarely
seen. For example, many consistent estimators admit high order stochastic
expansions, a natural question is how to quantify the lower bounds for the
uncertainty of these high order terms. Also, there is immense work on high order
efficient estimators, which are constructed from lower order efficient ones with
high order bias corrections. This induces extra uncertainty and raises the
question of how to quantify the least such variation. Assume expansions of
estimators in orders of we attempt to formulate the high
order version of the convolution representation of Hájek-Le Cam, and then
obtain high order information lower bound in cases of parametric, semiparametric
and in presence of nuisance parameters.
Keywords and phrases: convolution representation, high order expansion, information bound, nuisance parameter, regular estimator.
