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 νNRk, and the nuisance parameter ηHRm. Thus θ = (ν,η) ∈ N×HRk+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

  1. 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.

Other useful references

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