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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hájek–Le Cam convolution theorem ： ウィキペディア英語版 Hájek–Le Cam convolution theorem 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. == Theorem statement == Let ℘ = be a regular parametric model, and ''q''(''θ''): Θ → ℝ''m'' 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 ''q̇θ''. Define : $I\_^\; =\; \backslash dot(\backslash theta)I^(\backslash theta)\backslash dot(\backslash theta)\text{'}$ — the ''information bound'' for ''q'', : $\backslash psi\_\; =\; \backslash dot(\backslash theta)I^(\backslash theta)\backslash dot\backslash ell(\backslash theta)$ — the ''efficient influence function'' for ''q'', where ''I''(''θ'') is the Fisher information matrix for model ℘, $\backslash scriptstyle\backslash dot\backslash ell(\backslash theta)$ is the score function, and ′ denotes matrix transpose. Theorem . Suppose ''Tn'' is a uniformly (locally) regular estimator of the parameter ''q''. Then There exist independent random ''m''-vectors $\backslash scriptstyle\; Z\_\backslash theta\backslash ,\backslash sim\backslash ,\backslash mathcal(0,\backslash ,I^\_)$ and ''Δθ'' such that : $$ \sqrt(T_n - q(\theta)) \ \xrightarrow\ Z_\theta + \Delta_\theta, where ''d'' denotes convergence in distribution. More specifically, : $$ \begin \sqrt(T_n - q(\theta)) - \tfrac^n \psi_(x_i) \\ \tfrac^n \psi_(x_i) \end \ \xrightarrow\ \begin \Delta_\theta \\ Z_\theta \end. If the map ''θ'' → ''q̇θ'' is continuous, then the convergence in (A) holds uniformly on compact subsets of Θ. Moreover, in that case Δ''θ'' = 0 for all ''θ'' if and only if ''Tn'' is uniformly (locally) asymptotically linear with influence function ''ψ''''q''(''θ'') 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hájek–Le Cam convolution theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.