Estimating the mixing density of a mixture distribution remains an interesting problem in statistics literature. Using a stochastic approximation method, Newton and Zhang (1999) introduced a fast recursive algorithm for estimating the mixing density of a mixture. Under suitably chosen weights the stochastic approximation estimator converges to the true solution. In Tokdar et. al. (2009) the consistency of this recursive estimation method was established. However, the proof of consistency of the resulting estimator used independence among observations as an assumption. Here, we extend the investigation of performance of Newton's algorithm to several dependent scenarios. We first prove that the original algorithm under certain conditions remains consistent when the observations are arising form a weakly dependent process with fixed marginal with the target mixture as the marginal density. For some of the common dependent structures where the original algorithm is no longer consistent, we provide a modification of the algorithm that generates a consistent estimator.
翻译:在统计文献中,估计混合物分布的混合密度仍然是一个有趣的问题。使用随机近似法,牛顿和张(1999年)采用了快速递归算法来估计混合物的混合密度。在适当选择的重量下,随机近似估测仪与真正的解决方案汇合。在Tokdar等人(2009年)一案中,确定了这种循环估计方法的一致性。然而,由此得出的估计方法的一致性证据将观测结果的独立性作为假设。在这里,我们将对牛顿算法的绩效的调查扩大到若干依附情景。我们首先证明,当观测结果形成一个依赖性弱的过程,与目标混合物作为边际密度具有固定边缘性时,在某些条件下,原始算法仍然保持不变。对于一些原算法不再一致的常见依赖结构,我们提供了一种对算法的修改,以产生一致的估测值。