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 , and the nuisance parameter . Thus . Then we will say that $\scriptstyle\hat\nu_n$ is an adaptive estimator of ''ν'' in the presence of ''η'' if this estimator is regular, and efficient for each of the submodels
: $\mathcal_\nu(\eta_0) = \big\.$
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
:
I_(\theta) = \operatorname , z_\nu z_\eta' \,= 0 \quad \text\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'').

Example

Suppose $\scriptstyle\mathcal$ is the normal location-scale family:
: $\mathcal = \Big\.$
Then the usual estimator $\hat\mu\,=\,\bar$ is adaptive: we can estimate the mean equally well whether we know the variance or not.

Notes

Basic references

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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.

Estimator