Adaptive estimator explained

In statistics, an adaptive estimator is an estimator in a parametric or semiparametric model with nuisance parameters such that the presence of these nuisance parameters does not affect efficiency of estimation.

Definition

Formally, let parameter θ in a parametric model consists of two parts: the parameter of interest, and the nuisance parameter . Thus . Then we will say that $\backslash scriptstyle\backslash hat\backslash nu\_n$ is an adaptive estimator of ν in the presence of η if this estimator is regular, and efficient for each of the submodels

l{P} _{\nu(η} _{0)} = \ { P _{\theta:} \nu\in N, η=η _{0\}.} Adaptive estimator estimates the parameter of interest equally well regardless whether the value of the nuisance parameter is known or not.

The necessary condition for a regular parametric model to have an adaptive estimator is that

I _{\nuη} (\theta) = \operatorname{E}[ z _{\nu} z _{η'} ] = 0 forall \theta, where z_ν and z_η are components of the score function corresponding to parameters ν and η respectively, and thus I_νη is the top-right k×m block of the Fisher information matrix I(θ).

Example

Suppose

\scriptstyle l{P} is the normal location-scale family:l{P} = \ { f _{\theta(x)} = \tfrac{ 1 }{\sqrt{ 2 \pi}\sigma} e ^{- (1)/(2 \sigma ^{2}) (x - \mu) ^{2}} | \mu\in R , \sigma> 0 \ }. Then the usual estimator \hat\mu = \bar{x} is adaptive: we can estimate the mean equally well whether we know the variance or not.

Basic references

• Book: Bickel, Peter J. . Chris A.J. Klaassen . Ya’acov Ritov . Jon A. Wellner . Efficient and adaptive estimation for semiparametric models . Springer: New York . 1998 . 978-0-387-98473-5 . CITEREFBickel1998 .

Other useful references

• I. V. Blagouchine and E. Moreau: "Unbiased Adaptive Estimations of the Fourth-Order Cumulant for Real Random Zero-Mean Signal", IEEE Transactions on Signal Processing, vol. 57, no. 9, pp. 3330–3346, September 2009.