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| 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 [[independence (probability theory)|independent]] random variables, one of which is [[normal distribution|normal]] with asymptotic variance equal to the inverse of [[Fisher information]], and the other having arbitrary distribution.
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| 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 [[parametric model#Regular parametric model|regular parametric model]]s. | |
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| The theorem is named after [[Jaroslav Hájek]] and [[Lucien Le Cam]].
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| == Theorem statement ==
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| Let ℘ = {''P<sub>θ</sub>'' | ''θ'' ∈ Θ ⊂ ℝ<sup>''k''</sup>} be a [[parametric model#Regular parametric model|regular parametric model]], and ''q''(''θ''): Θ → ℝ<sup>''m''</sup> 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 denotes as ''q̇<sub>θ</sub>''. Define
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| : <math> I_{q(\theta)}^{-1} = \dot{q}(\theta)I^{-1}(\theta)\dot{q}(\theta)'</math> — the ''information bound'' for ''q'',
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| : <math> \psi_{q(\theta)} = \dot{q}(\theta)I^{-1}(\theta)\dot\ell(\theta)</math> — the ''efficient influence function'' for ''q'',
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| where ''I''(''θ'') is the [[Fisher information]] matrix for model ℘, <math>\scriptstyle\dot\ell(\theta)</math> is the [[score (statistics)|score function]], and ′ denotes [[matrix transpose]].
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| <br/>
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| '''Theorem''' {{harv|Bickel|1998|loc=Th.2.3.1}}. Suppose ''T<sub>n</sub>'' is a uniformly (locally) [[regular estimator]] of the parameter ''q''. Then | |
| <ol type=A> | |
| <li> There exist independent random ''m''-vectors <math>\scriptstyle Z_\theta\,\sim\,\mathcal{N}(0,\,I^{-1}_{q(\theta)})</math> and ''Δ<sub>θ</sub>'' such that | |
| : <math> | |
| \sqrt{n}(T_n - q(\theta)) \ \xrightarrow{d}\ Z_\theta + \Delta_\theta,
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| </math>
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| where <sup>''d''</sup> denotes [[convergence in distribution]]. More specifically,
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| : <math>
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| \begin{pmatrix}
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| \sqrt{n}(T_n - q(\theta)) - \tfrac{1}{\sqrt{n}} \sum_{i=1}^n \psi_{q(\theta)}(x_i) \\
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| \tfrac{1}{\sqrt{n}} \sum_{i=1}^n \psi_{q(\theta)}(x_i)
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| \end{pmatrix}
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| \ \xrightarrow{d}\
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| \begin{pmatrix}
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| \Delta_\theta \\
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| Z_\theta
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| \end{pmatrix}.
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| </math>
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| <li> If the map ''θ'' → ''q̇<sub>θ</sub>'' is continuous, then the convergence in (A) holds uniformly on compact subsets of Θ. Moreover, in that case Δ<sub>''θ''</sub> = 0 for all ''θ'' if and only if ''T<sub>n</sub>'' is uniformly (locally) asymptotically linear with influence function ''ψ''<sub>''q''(''θ'')</sub>
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| </ol>
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| ==References==
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| * {{cite book
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| | authors = Bickel, Peter J.; Klaassen, Chris A.J.; Ritov, Ya’acov; Wellner Jon A.
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| | title = Efficient and adaptive estimation for semiparametric models
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| | publisher = Springer
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| | location = New York
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| | year = 1998
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| | isbn = 0-387-98473-9
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| | ref = CITEREFBickel1998
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| }}
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| {{reflist}}
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| {{DEFAULTSORT:Hajek-Le Cam Convolution Theorem}}
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| [[Category:Statistical theorems]]
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