Adaptive estimator
# Adaptive estimator

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 ν ∈ N ⊆ $R^k,$ and the nuisance parameter η ∈ H ⊆ $R^m.$ Thus θ = (ν,η) ∈ N×H ⊆ $R^{k+m}.$ Then we will say that $\scriptstyle {\hat {\nu }}_{n}$ is an adaptive estimator of ν in the presence of η if this estimator is regular, and efficient for each of the submodels

    ${\mathcal {P}}_{\nu }(\eta _{0})={\big \P_{\theta }:\nu \in N,\,\eta =\eta _{0}{\big \}}.$

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

    $I_{\nu \eta }(\theta )=\operatorname {E} [\,z_{\nu }z_{\eta }'\,]=0\quad {\text{for all }}\theta ,$

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 $\scriptstyle {\mathcal {P}}$ is the normal location-scale family:

    ${\mathcal {P}}={\Big \\ f_{\theta }(x)={\tfrac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {1}{2\sigma ^{2}}}(x-\mu )^{2}}\ {\Big |}\ \mu \in \mathbb {R} ,\sigma >0\ {\Big \}}.$

Then the usual estimator ${\hat {\mu }}\,=\,{\bar {x}}$ is adaptive: we can estimate the mean equally well whether we know the variance or not.

