Adaptive estimator

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 ${\displaystyle \scriptstyle {\hat {\nu }}_{n}}$ is an adaptive estimator of ν in the presence of η if this estimator is regular, and efficient for each of the submodels^[1]

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

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

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

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).

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

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

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