# How Many Iterations are Sufficient for Semiparametric Estimation - Mathematics > Statistics Theory

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Abstract: A common practice in obtaining a semiparametric efficient estimate is throughiteratively maximizing the penalized log-likelihood w.r.t. its Euclideanparameter and functional nuisance parameter via Newton-Raphson algorithm. Thepurpose of this paper is to provide a formula in calculating the minimal numberof iterations $k^\ast$ needed to produce an efficient estimate$\hat\theta n^{k^\ast}$ from a theoretical point of view. We discover thata $k^\ast$ depends on the convergence rates of the initial estimate andnuisance estimate; b more than $k^\ast$ iterations, i.e., $k$, will onlyimprove the higher order asymptotic efficiency of $\hat\theta n^{k}$; c$k^\ast$ iterations are also sufficient for recovering the estimation sparsityin high dimensional data. These general conclusions hold, in particular, whenthe nuisance parameter is not estimable at root-n rate, and apply tosemiparametric models estimated under various regularizations, e.g., kernel orpenalized estimation. This paper provides a first general theoreticaljustification for the -one-two-step iteration- phenomena observed in theliterature, and may be useful in reducing the bootstrap computational cost forthe semiparametric models.

Author: ** Guang Cheng**

Source: https://arxiv.org/