Singularity structures and impacts on parameter estimation in finite mixtures of distributions
Nhat Ho, XuanLong Nguyen
备注：
87 pages. This version has improved introduction and expanded discussion of related work. An abridged version is to appear on SIAM Journal on Mathematics of Data Science
Singularities of a statistical model are the elements of the model's parameter space which make the corresponding Fisher information matrix degenerate. These are the points for which estimation techniques such as the maximum likelihood estimator and standard Bayesian procedures do not admit the root-$n$ parametric rate of convergence. We propose a general framework for the identification of singularity structures of the parameter space of finite mixtures, and study the impacts of the singularity structures on minimax lower bounds and rates of convergence for the maximum likelihood estimator over a compact parameter space. Our study makes explicit the deep links between model singularities, parameter estimation convergence rates and minimax lower bounds, and the algebraic geometry of the parameter space for mixtures of continuous distributions. The theory is applied to establish concrete convergence rates of parameter estimation for finite mixture of skew-normal distributions. This rich and increasingly popular mixture model is shown to exhibit a remarkably complex range of asymptotic behaviors which have not been hitherto reported in the literature.
A Bayesian Mixture Modelling of Stop Signal Reaction Time Distributions
Hans Colonius（1990）和Eric-Jan Wagenmakers及其同事（2012）分别使用非参数方法和贝叶斯参数方法对停止信号任务（SST）中单个停止信号反应时间（SSRT）的分布进行了建模，该分布作为不可观察停止过程的延迟测量。这些方法假设SST试验中的前一试验类型（进行/停止）对SSRT分布估计的影响相等，而不解决违反假设的情况。本研究提出了所需的模型，考虑两个状态的混合模型的SSRT分布。然后比较了贝叶斯参数的单一SSRT和混合SSRT分布在通常的随机顺序在个人和人口水平下的前高斯分布格式。结果表明，与单一SSRT分布相比，混合SSRT分布具有更大的多样性、更大的正偏性、更高的尖峰性和更大的随机序。结果中差异的大小还取决于混合SSRT分布中权重的选择。这项研究证实，混合SSRT指数作为一个常数或分布显着大于其单一的SSRT对应的相关顺序。这为SSRT估计提供了至关重要的改进。
