In statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, an adaptive estimator is an estimator

Estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....

 in a parametric

Parametric model

In statistics, a parametric model or parametric family or finite-dimensional model is a family of distributions that can be described using a finite number of parameters...

 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 $\backslash scriptstyle\backslash hat\backslash nu\_n$ is an adaptive estimator of ν in the presence of η if this estimator is regular, and efficient for each of the submodels

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 z_ν and z_η are components of the score function

Score function

The term score function may refer to:* Scoring rule, in decision theory, a measure of one's performance when making decisions under uncertainty* Score , the derivative of the log-likelihood function with respect to the parameter...

 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 is the normal location-scale family

Location-scale family

In probability theory, especially as that field is used in statistics, a location-scale family is a family of univariate probability distributions parametrized by a location parameter and a non-negative scale parameter; if X is any random variable whose probability distribution belongs to such a...

:

Then the usual estimator is adaptive: we can estimate the mean equally well whether we know the variance or not.

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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