Adaptive estimator - Alchetron, The Free Social Encyclopedia

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 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

P ν ( η 0 ) = { P θ : ν ∈ N , η = η 0 } .

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 ν η ( θ ) = E ⁡ [ z ν z η ′ ] = 0 for all θ ,

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 P is the normal location-scale family:

P = { f θ ( x ) = 1 2 π σ e − 1 2 σ 2 ( 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.

References

Adaptive estimator Wikipedia

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Contents

Definition

Example

References