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 ⊆ Rk, and the nuisance parameter η ∈ H ⊆ Rm. Thus θ = (ν,η) ∈ N×H ⊆ Rk+m. Then we will say that is an adaptive estimator of ν in the presence of η if this estimator is regular, and efficient for each of the submodels[1]
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
where zν and zη are components of the score function corresponding to parameters ν and η respectively, and thus Iνη is the top-right k×m block of the Fisher information matrix I(θ).
Example
Suppose is the normal location-scale family:
Then the usual estimator is adaptive: we can estimate the mean equally well whether we know the variance or not.
Notes
- ↑ Bickel 1998, Definition 2.4.1
Basic references
- Bickel, Peter J.; Chris A.J. Klaassen; Ya’acov Ritov; Jon A. Wellner (1998). Efficient and adaptive estimation for semiparametric models. Springer: New York. ISBN 978-0-387-98473-5.