Adaptive estimator

Formally, let parameter θ in a parametric model consists of two parts: the parameter of interest {{nowrap|ν ∈ N ⊆ R^k}}, and the nuisance parameter {{nowrap|η ∈ H ⊆ R^m}}. Thus {{nowrap|1=θ = (ν,η) ∈ N×H ⊆ R^k+m}}. 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{{harvnb|Bickel|1998|loc=Definition 2.4.1}}

: $$

\mathcal{P}_\nu(\eta_0) = \big\{ P_\theta: \nu\in N,\, \eta=\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_{\nu\eta}(\theta) = \operatorname{E}[\, z_\nu z_\eta' \,] = 0 \quad \text{for all }\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).

Suppose $\backslash scriptstyle\backslash mathcal\{P\}$ is the normal location-scale family:

: $$

\mathcal{P} = \Big\{\ f_\theta(x) = \tfrac{1}{\sqrt{2\pi}\sigma} e^{ -\frac{1}{2\sigma^2}(x-\mu)^2 }\ \Big|\ \mu\in\mathbb{R}, \sigma>0 \ \Big\}.

Then the usual estimator $\backslash hat\backslash mu\backslash ,=\backslash ,\backslash bar\{x\}$ is adaptive: we can estimate the mean equally well whether we know the variance or not.

{{reflist}}

{{refbegin}}

• {{cite book

| author = Bickel, Peter J.

| author2 = Chris A.J. Klaassen

| author3 = Ya’acov Ritov

| author4 = Jon A. Wellner

| title = Efficient and adaptive estimation for semiparametric models

| publisher = Springer: New York

| year = 1998

| isbn = 978-0-387-98473-5

| ref = CITEREFBickel1998

}}

{{refend}}

• [https://ieeexplore.ieee.org/document/4838921/;jsessionid=6FED64C70D5633F166D3346F6833703C?isnumber=5200771&arnumber=4838921&count=38&index=2 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.]

Category:Estimator