一种用于似然最大化的类牛顿算法：稳健方差评分算法
A Newton-Like Algorithm for Likelihood Maximization: The Robust-Variance Scoring Algorithm
AI总结：
提出稳健方差评分（RVS）算法，用得分方差估计替代牛顿-拉夫森算法中的海森矩阵，减少计算量，并基于近似误差与统计误差之比设计收敛准则，模拟表明RVS比Marquardt算法更快且覆盖率满意。
摘要：
本文研究了一种已被多位作者使用但尚未被彻底研究的类牛顿方法。我们称之为稳健方差评分（RVS）算法，因为我们考虑的主要版本用矩阵$G$替换了牛顿-拉夫森算法中使用的对数似然的负海森矩阵，该矩阵是在真实概率下得分的方差估计，仅使用个体得分。因此，该算法的一次迭代所需的计算量远少于牛顿-拉夫森算法的一次迭代。此外，这种得分方差的估计在最大值处估计信息矩阵。我们还研究了一个收敛准则，该准则具有估计近似误差与统计误差之比的良好解释；因此它可以用于停止迭代过程，无论问题如何。模拟研究证实，RVS算法比Marquardt算法（牛顿-拉夫森算法的稳健版本）更快；这是因为RVS算法所需的迭代次数仅略大于Marquardt算法，而每次迭代的计算时间却短得多。此外，使用矩阵$G$的覆盖率也是令人满意的。
英文摘要：
This article studies a Newton-like method already used by several authors but which has not been thouroughly studied yet. We call it the robust-variance scoring (RVS) algorithm because the main version of the algorithm that we consider replaces minus the Hessian of the loglikelihood used in the Newton-Raphson algorithm by a matrix $G$ which is an estimate of the variance of the score under the true probability, which uses only the individual scores. Thus an iteration of this algorithm requires much less computations than an iteration of the Newton-Raphson algorithm. Moreover this estimate of the variance of the score estimates the information matrix at maximum. We have also studied a convergence criterion which has the nice interpretation of estimating the ratio of the approximation error over the statistical error; thus it can be used for stopping the iterative process whatever the problem. A simulation study confirms that the RVS algorithm is faster than the Marquardt algorithm (a robust version of the Newton-Raphson algorithm); this happens because the number of iterations needed by the RVS algorithm is barely larger than that needed by the Marquardt algorithm while the computation time for each iteration is much shorter. Also the coverage rates using the matrix $G$ are satisfactory.
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