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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 Two-step M-estimators involving MLE ： ウィキペディア英語版 Two-step M-estimators involving MLE Two-step M-estimator involving Maximum Likelihood Estimator is a special case of general two-step M-estimator. Thus, consistency and asymptotic normality of the estimator follows from the general result on two-step M-estimators. Yet, when the first step estimation is MLE, under some assumptions, two-step M-estimator is more efficient (has smaller asymptotic variance ) than M-estimator with known first-step parameter 〔Wooldridge, J.M., Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.〕 Let i=1n be a random sample and the second-step M-estimator $\backslash widehat$ is the following: $\backslash widehat$ ≔ $$ \undersetm(\,v_i,w_i,z_i: \theta\,,\widehat) where $\backslash widehat$ is the parameter estimated by ML procedure in the first step. For the MLE, $\backslash widehat$ ≔ $$ \underset\log f(v_ : z_ , \gamma) where ''f'' is the conditional density of ''V'' given ''Z''. Now, suppose that given ''Z, V'' is conditionally independent of ''W''. This assumption is called conditional independence assumption or selection on observables 〔Heckman, J.J., and R. Robb, 1985, Alternative Methods for Evaluating the Impact of Interventions: An Overview, Journal of Econometrics, 30, 239-267.〕 〔. Intuitively, this condition means that Z is a good predictor of V so that once conditioned on ''Z, V'' has no systematic dependence on ''W''. Under the conditional independence assumption, the asymptotic variance of the two-step estimator is: ''E( s(θ0,γ0) )-1 E()g(θ0,γ0 )' )E( s(θ0,γ0) )-1'' where ''g(θ,γ) ≔ s(θ,γ)-E(s(θ , γ) ∇γ d(γ)' )E(d(γ) ∇γ d(γ)' )-1 d(γ), s(θ,γ) ≔ ∇θ m(V, W, Z: θ, γ) , d(γ) ≔ ∇γ log f (V : Z, γ)'', and ∇ represents partial derivative with respect to a row vector. In the case where ''γ0'' is known, the asymptotic variance is ''E( s(θ0,γ0) )-1 E()s(θ0,γ0 )' )E( s(θ0,γ0) )-1 and therefore, unless E(s(θ, γ) ∇γ d(γ)' )=0'', the two-step M-estimator is more efficient than the usual M-estimator. This fact suggests that even when ''γ0'' is known a priori, there is efficiency gain by estimating ''γ'' by MLE. An application of this result can be found, for example, in treatment effect estimation 〔. 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「Two-step M-estimators involving MLE」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.