In statistics, the Hájek–Le Cam convolution theorem states that any regular estimator in a parametric model is asymptotically equivalent to a sum of two independent random variables, one of which is normal with asymptotic variance equal to the inverse of Fisher information, and the other having arbitrary distribution.
The obvious corollary from this theorem is that the “best” among regular estimators are those with the second component identically equal to zero. Such estimators are called efficient and are known to always exist for regular parametric models.
The theorem is named after Jaroslav Hájek and Lucien Le Cam.
Let ℘ = be a regular parametric model, and q(θ): Θ → ℝm be a parameter in this model (typically a parameter is just one of the components of vector θ). Assume that function q is differentiable on Θ, with the m × k matrix of derivatives denoted as q̇θ. Define
| -1 | |
| I | |
| q(\theta) |
=
| q |
(\theta)I-1(\theta)
| q |
(\theta)'
\psiq(\theta)=
| q |
(\theta)I-1(\theta)
| \ell(\theta) |
where I(θ) is the Fisher information matrix for model ℘,
| \scriptstyle\ell(\theta) |
Theorem . Suppose Tn is a uniformly (locally) regular estimator of the parameter q. Then
\scriptstyle
| -1 | |
| Z | |
| q(\theta) |
)
\sqrt{n}(Tn-q(\theta)) \xrightarrow{d} Z\theta+\Delta\theta,
\begin{pmatrix} \sqrt{n}(Tn-q(\theta))-\tfrac{1}{\sqrt{n}}
| n | |
| \sum | |
| i=1 |
\psiq(\theta)(xi)\\ \tfrac{1}{\sqrt{n}}
| n | |
| \sum | |
| i=1 |
\psiq(\theta)(xi) \end{pmatrix} \xrightarrow{d} \begin{pmatrix} \Delta\theta\\ Z\theta \end{pmatrix}.