Adaptive Estimator

An Adaptive Estimator is an efficient estimator for a parametric or semiparametric model.

• See: Estimator, Parametric Model, Nuisance Parameter.

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References

2016

• (Wikipedia, 2016) ⇒ http://www.wikiwand.com/en/Adaptive_estimator Retrieved 2016-07-10
  • 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 [math]\displaystyle{ \scriptstyle\hat\nu_n }[/math] is an adaptive estimator of ν in the presence of η if this estimator is regular, and efficient for each of the submodels

[math]\displaystyle{ \mathcal{P}_\nu(\eta_0) = \big\{ P_\theta: \nu\in N,\, \eta=\eta_0\big\}. }[/math]

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

[math]\displaystyle{ I_{\nu\eta}(\theta) = \operatorname{E}[\, z_\nu z_\eta' \,] = 0 \quad \text{for all }\theta, }[/math]

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).

1992

• (Steigerwald,1992) ⇒ Steigerwald, D. G. (1992). Adaptive estimation in time series regression models. Journal of Econometrics, 54(1-3), 251-275. http://www.econ.ucsb.edu/papers/wp04-06.pdf
  • An adaptive estimator is an efficient estimator for a model that is only partially specified. For example, consider estimating a parameter that describes a sample of observations drawn from a distribution F. One natural question is: Is it possible that an estimator of the parameter constructed without knowledge of F could be as efficient (asymptotically) as any well-behaved estimator that relies on knowledge of F? For some problems the answer is yes, and the estimator that is efficient is termed an adaptive estimator.

Consider the familiar scalar linear regression model

[math]\displaystyle{ Y_t = \beta_0 + \beta_1X_t + U_t }[/math]

where the regressor is exogenous and {[math]\displaystyle{ Ut }[/math]} is a sequence of n independent and identically distributed random variables with distribution F. The parameter vector [math]\displaystyle{ \beta = (\beta_0, \beta_1) }[/math] 0 is often of interest rather than the distribution of the error, F. If we assume that F is described by a parameter vector [math]\displaystyle{ \alpha }[/math] (that is, we parameterize the distribution), then the resultant (maximum likelihood or ML) estimator of [math]\displaystyle{ \beta }[/math] is parametric. If we assume only that F belongs to a family of distributions, then the resultant estimator of [math]\displaystyle{ \beta }[/math] is semiparametric. Because the OLS estimator does not require that we parameterize F, the OLS estimator is semiparametric. If the population error distribution is Gaussian, we know that the OLS estimator is equivalent to the ML estimator, and so is efficient.

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