- Source: 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 ν ∈ N ⊆ Rk, and the nuisance parameter η ∈ H ⊆ Rm. Thus θ = (ν,η) ∈ N×H ⊆ Rk+m. Then we will say that
ν
^
n
{\displaystyle \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
P
ν
(
η
0
)
=
{
P
θ
:
ν
∈
N
,
η
=
η
0
}
.
{\displaystyle {\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
ν
η
(
θ
)
=
E
[
z
ν
z
η
′
]
=
0
for all
θ
,
{\displaystyle I_{\nu \eta }(\theta )=\operatorname {E} [\,z_{\nu }z_{\eta }'\,]=0\quad {\text{for all }}\theta ,}
where zν and zη are components of the score function corresponding to parameters ν and η respectively, and thus Iνη is the top-right k×m block of the Fisher information matrix I(θ).
Example
Suppose
P
{\displaystyle \scriptstyle {\mathcal {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
}
.
{\displaystyle {\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
μ
^
=
x
¯
{\displaystyle {\hat {\mu }}\,=\,{\bar {x}}}
is adaptive: we can estimate the mean equally well whether we know the variance or not.
Notes
Basic references
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.
Kata Kunci Pencarian:
- Statistika
- Ilmu aktuaria
- Statistika matematika
- Variabel acak
- Model generatif
- Efek pengacau
- Eksperimen semu
- Adaptive estimator
- Minimum-variance unbiased estimator
- Bias of an estimator
- Efficiency (statistics)
- M-estimator
- List of statistics articles
- Rao–Blackwell theorem
- Median
- Estimation theory
- Maximum likelihood estimation
