Using stochastic gradient search and the optimal filter derivative, it is possible to perform recursive maximum likelihood estimation in a non-linear state-space model. As the optimal filter and its derivative are analytically intractable for such a model, they need to be approximated numerically. In Poyiadjis et al. (G. Poyiadjis, A. Doucet, and S. S. Singh, Biometrika, vol. 98, no. 1, pp. 65–80, 2011), a recursive maximum likelihood algorithm based on a particle approximation to the optimal filter derivative has been proposed and studied through numerical simulations. This algorithm and its asymptotic behavior are here analyzed theoretically. Under regularity conditions, we show that the algorithm accurately estimates maxima of the underlying log-likelihood rate when the number of particles is sufficiently large. We also provide qualitative upper bounds on the estimation error in terms of the number of particles.

中文翻译：

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递归粒子最大似然估计的渐近性质

使用随机梯度搜索和最佳滤波器导数，可以在非线性状态空间模型中执行递归最大似然估计。由于对于这种模型，最优滤波器及其导数在分析上难以处理，因此需要在数值上进行近似。在波亚吉斯等。（G. Poyiadjis，A.Doucet和SS Singh，Biometrika，第98卷，第1期，第65-80页，2011年），提出了一种基于粒子近似于最优滤波器导数的递归最大似然算法。并通过数值模拟进行了研究。从理论上分析了该算法及其渐近行为。在规则性条件下，我们表明，当粒子数量足够大时，该算法可以准确地估计潜在对数似然率的最大值。我们还根据粒子数量提供了估计误差的定性上限。