One-Step Targeted Minimum Loss-based Estimation Based on Universal Least Favorable One-Dimensional Submodels.
One-Step Targeted Minimum Loss-based Estimation Based on Universal Least Favorable One-Dimensional Submodels.
复制标题
基于通用最不利一维子模型的一步目标最小损失估计。
DOI：
10.1515/ijb-2015-0054
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发表时间：
2016-05-01
期刊：
影响因子：
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通讯作者：
Gruber S
中科院分区：
文献类型：
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作者：
van der Laan M;Gruber S
考虑一项研究，其中观察 n 个独立且同分布的随机变量，已知其概率分布是特定统计模型的元素，并且关注该数据概率分布的特定实值路径可微目标参数的估计。目标最大似然估计器（TMLE）是一种渐进有效的替代估计器，通过在初始估计器零波动时，跨越有效影响曲线的初始估计器构造所谓的最不利参数子模型，并迭代地最大化相应的参数似然直到不再发生更新，此时更新的初始估计器求解所谓的有效影响曲线方程。在本文中，我们构建了一个一维通用最不利子模型，TMLE只需要一步，因此需要最少的额外数据拟合来实现求解有效影响曲线方程的目标。我们根据目标最小损失估计的需要，通过数据分布的相关部分将这些概括为通用最不利子模型。最后，值得注意的是，给定多维目标参数，我们开发了一种通用规范一维子模型，使得单步 TMLE 仅最大化单变量参数的对数似然，从而求解多变量有效影响曲线方程。这使我们能够通过初始估计器构建基于一维参数子模型的单步 TMLE，从而求解任何多元所需的估计方程组。
Consider a study in which one observes n independent and identically distributed random variables whose probability distribution is known to be an element of a particular statistical model, and one is concerned with estimation of a particular real valued pathwise differentiable target parameter of this data probability distribution. The targeted maximum likelihood estimator (TMLE) is an asymptotically efficient substitution estimator obtained by constructing a so called least favorable parametric submodel through an initial estimator with score, at zero fluctuation of the initial estimator, that spans the efficient influence curve, and iteratively maximizing the corresponding parametric likelihood till no more updates occur, at which point the updated initial estimator solves the so called efficient influence curve equation. In this article we construct a one-dimensional universal least favorable submodel for which the TMLE only takes one step, and thereby requires minimal extra data fitting to achieve its goal of solving the efficient influence curve equation. We generalize these to universal least favorable submodels through the relevant part of the data distribution as required for targeted minimum loss-based estimation. Finally, remarkably, given a multidimensional target parameter, we develop a universal canonical one-dimensional submodel such that the one-step TMLE, only maximizing the log-likelihood over a univariate parameter, solves the multivariate efficient influence curve equation. This allows us to construct a one-step TMLE based on a one-dimensional parametric submodel through the initial estimator, that solves any multivariate desired set of estimating equations.