捕获再捕获模型下的半参数惩罚经验似然估计理论及EM算法

在生态学、流行病学、社会学等许多领域, 确定总体大小对于生态保护、疫情防控、人口结构优化等问题具有重要意义。估计总体大小最常用的方法是捕获再捕获抽样。针对捕获再捕获数据，文献中对总体大小的估计方法大都基于逆概率加权或完全似然。当总体规模不大时，总体大小的Horvitz-Thompson型估计量和最大似然估计量可能不够稳健; Wald型置信区间的覆盖概率可能严重低于名义置信水平，似然比置信上限可能无界；最大化似然函数的数值算法不够稳定。本项目将在参数捕获概率模型假设下对总体大小开发一套半参数惩罚经验似然估计理论以及高效稳定的EM算法，将其拓展到单调凸的非参数捕获概率模型、协变量存在缺失和测量误差以及被捕次数1过多的捕获再捕获复杂数据，并建立相应的大样本理论和EM算法的收敛性。本项目的研究成果不仅对完善半参数统计方法具有重要意义，而且对生态学、社会学等相关学科的发展具有极大的促进作用。

In ecology, epidemiology, sociology and many other fields, determining the size of a generic population is of great importance for ecological conservation, epidemic prevention and control, population structure optimization and other issues. Capture-recapture sampling methods are the most widely used methods to estimate population sizes. From capture-recapture data, the estimation methods for population sizes in the literature are mostly based on inverse probability weighting or full likelihood. When the population size is small, the Horvitz-Thompson type estimator and maximum likelihood estimator may not be robust enough. The coverage probability of Wald confidence intervals may be lower than the nominal confidence level and the upper limit of likelihood ratio confidence interval may be unbounded. In addition, the numerical algorithms of maximizing likelihood functions may be not stable enough. In this project, a set of semi-parametric penalized empirical likelihood estimation theory and efficient and stable EM algorithms are developed for estimating the population sizes under parametric capture probability models. They are further extended to nonparametric capture probability models with monotonic and convex constraints, and complicated capture-capture data where one captures are inflated and covariates are subject to missingness and measurement errors. What’s more, in this project we investigate the corresponding large-sample properties and the convergence property of EM algorithms. The research results of this project are not only of great significance to improve the semi-parametric statistical methods, but also have great promotion to the developments of ecology, sociology, and other related disciplines.