Adaptive estimator

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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.

## Contents

* 1 Definition
* 2 Example
* 3 Notes
* 4 Basic references
* 5 Other useful references

## 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 ν^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]

𝒫ν(η0)={Pθ:ν∈N,η=η0}.

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

Iνη(θ)=E⁡[zνzη′]=0for all θ,

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 𝒫 is the normal location-scale family:

𝒫={ fθ(x)=12πσe−12σ2(x−μ)2 | μ∈ℝ,σ>0 }.

Then the usual estimator μ^=x¯ 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

* 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.

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