We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. The model considered in the paper is very general as we do not impose any preliminary restrictions on the probabilistic nature of the observation process and cover a wide class of nonlinear recursive procedures. In this paper we study asymptotic behaviour of the recursive estimators. The results of the paper can be used to determine the form of a recursive procedure which is expected to have the same asymptotic properties as the corresponding non-recursive one defined as a solution of the corresponding estimating equation.
Stochastic optimisation in Riemannian manifolds, especially the Riemannian stochastic gradient method, has attracted much recent attention. The present work applies stochastic optimisation to the task of recursive estimation of a statistical parameter which belongs to a Riemannian manifold. Roughly, this task amounts to stochastic minimisation of a statistical divergence function. The following problem is considered : how to obtain fast, asymptotically efficient, recursive estimates, using a Riemannian stochastic optimisation algorithm with decreasing step sizes? In solving this problem, sever…
In this work, we study a new recursive stochastic algorithm for the joint estimation of quantile and superquantile of an unknown distribution. The novelty of this algorithm is to use the Cesaro averaging of the quantile estimation inside the recursive approximation of the superquantile. We provide some sharp non-asymptotic bounds on the quadratic risk of the superquantile estimator for different step size sequences. We also prove new non-asymptotic $L^p$-controls on the Robbins Monro algorithm for quantile estimation and its averaged version. Finally, we derive a central limit theorem of our j…
In this paper, we consider the alleviation of the boundary problem when the probability density function has bounded support. We apply Robbins-Monro's algorithm and Bernstein polynomials to construct a recursive density estimator. We study the asymptotic properties of the proposed recursive estimator. We then compared our proposed recursive estimator with many others estimators. Finally, we confirm our theoretical result through a simulation study and then using two real datasets.
Using stochastic gradient search and the optimal filter derivative, it is possible to perform recursive (i.e., online) 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, Doucet and Singh, Biometrika 2018], a recursive maximum likelihood algorithm based on a particle approximation to the optimal filter derivative has been proposed and studied through numerical simulations. Here, this algorithm and its asymptotic behavior are analyzed theo…
Grid Based Nonlinear Filtering Revisited: Recursive Estimation & Asymptotic Optimality
我们回顾了一般马尔可夫过程在离散时间，部分观察到的条件高斯噪声的网格为基础的递归近似滤波的发展。所考虑的基于网格的滤波器依赖于两种类型的状态量化：\textit{Markovian}类型和\textit{marginal}类型。我们提出了一组新的，宽松的充分条件，确保强大的和充分的特点，这些过滤器的路径收敛到各自的MMSE状态估计器。特别是，对于边际状态量化，我们引入了\textit{conditional regularity of stochastic kernels}的概念，据我们所知，这构成了最宽松的条件下提出的，在此条件下，保证了各自的基于网格的过滤器的渐近最优性。此外，我们扩展我们的收敛结果，包括过滤的有界和连续的泛函的状态，以及递归近似状态预测。对于马尔可夫和边际量化，各自基于网格的滤波器的整个开发更多地依赖于线性代数技术，而不是测量理论参数，使得演示文稿相当短，技术上更简单。
The problem of parameter estimation by observations of inhomogeneous Poisson processes is considered. The method of moments estimator is studied and its stochastic expansion is obtained. This stochastic expansion is then used to obtain the expansion of the moments of this estimator and the expansion of the distribution function. The stochastic expansion, expansion of the moments and the expansion of distribution function are non asymptotic in nature. Several examples are considered.
Asymptotic expansion is presented for an estimator of the Hurst coefficient of a fractional Brownian motion. For this, a recently developed theory of asymptotic expansion of the distribution of Wiener functionals is applied. The effects of the asymptotic expansion are demonstrated by numerical studies.
