标题： 半参数估计需要多少次迭代才足够？ 显示英文标题

标题： How Many Iterations are Sufficient for Semiparametric Estimation?

摘要： 在获得半参数有效估计的一种常见做法是通过牛顿-拉夫森算法，对欧几里得参数和函数扰动参数进行迭代最大化（惩罚）对数似然。本文的目的是从理论角度提供一个公式，用于计算从理论上产生有效估计$\hat\theta_n^{(k^\ast)}$所需的最小迭代次数$k^\ast$。我们发现，(a)$k^\ast$依赖于初始估计和扰动估计的收敛速度；(b) 超过$k^\ast$次迭代，即$k$，只会提高$\hat\theta_n^{(k)}$的高阶渐近效率；(c)$k^\ast$次迭代也足以恢复高维数据中的估计稀疏性。这些一般结论在扰动参数不能以根号n速率估计时尤其成立，并适用于在各种正则化下估计的半参数模型，例如核估计或惩罚估计。本文为文献中观察到的“一步/两步迭代”现象提供了第一个一般的理论证明，并可能在减少半参数模型的Bootstrap计算成本方面有所帮助。 显示英文摘要

摘要： A common practice in obtaining a semiparametric efficient estimate is through iteratively maximizing the (penalized) log-likelihood w.r.t. its Euclidean parameter and functional nuisance parameter via Newton-Raphson algorithm. The purpose of this paper is to provide a formula in calculating the minimal number of iterations $k^\ast$ needed to produce an efficient estimate $\hat\theta_n^{(k^\ast)}$ from a theoretical point of view. We discover that (a) $k^\ast$ depends on the convergence rates of the initial estimate and nuisance estimate; (b) more than $k^\ast$ iterations, i.e., $k$, will only improve the higher order asymptotic efficiency of $\hat\theta_n^{(k)}$; (c) $k^\ast$ iterations are also sufficient for recovering the estimation sparsity in high dimensional data. These general conclusions hold, in particular, when the nuisance parameter is not estimable at root-n rate, and apply to semiparametric models estimated under various regularizations, e.g., kernel or penalized estimation. This paper provides a first general theoretical justification for the "one-/two-step iteration" phenomena observed in the literature, and may be useful in reducing the bootstrap computational cost for the semiparametric models.