Novel statistical methods for biased sampling problems

在许多科学学科中，如生物学、生态学、渔业研究和社会科学，都发现了有偏抽样。当收集到的数据的分布与目标人群的分布不同时，就会出现这种情况。例如，在捕获-再捕获实验中，较大的动物更有可能被捕获，因此观察到的数据是目标种群的有偏差样本。在不可忽略的缺失数据问题中，响应概率取决于丢失的研究变量；因此，完全观察到的数据的这个变量的分布与总体的分布不同，即使有协变量的条件。由于有偏抽样问题的固有性质，有效的统计推断是具有挑战性的。通过大量的文献回顾，我们发现现有的方法要么依赖于强参数假设，要么在参数估计中存在算法不稳定和效率损失。本研究计划旨在开发新的方法来解决有偏抽样问题，重点关注从捕获-再捕获数据和不可忽略的缺失数据中推断问题。第一个主题将考虑用捕获-再捕获数据估计封闭种群的丰度。首先，我们提出通过具有形状约束的广义加性模型来建模捕获概率，例如每个加性分量的单调性。其次，我们将开发一种惩罚经验似然（EL）方法来找到丰度的点估计和置信区间（CI）；这是一种有效的方法，可以防止对丰度进行虚假的大估计。第三，我们将设计一个有效且快速的算法来计算上述的点估计和CI。第二个主题将考虑不可忽视的数据丢失问题。我们将通过半参数Box-Cox转换模型对完全观测数据的协变量条件进行研究变量建模。我们将分两步进行。在第一步中，我们将开发一种极大二项似然方法来分析半参数Box-Cox变换模型。在第二步中，我们将把该方法与EL方法结合起来，为响应概率模型中研究变量和未知参数的均值建立有效的估计量。本提案中的项目将开发新的无调优参数统计方法，以解决具有捕获-再捕获数据和不可忽略的缺失数据问题的丰度估计中的重要和具有挑战性的问题。这些方法在理论上是可靠的，并在可公开访问的R包中实现。这个研究项目将有利于非参数似然方法和形状约束推理的理论和方法研究。其成果将适用于加拿大在卫生科学、经济学、野生动物管理和社会科学方面出现的一系列科学问题。

Biased sampling has been identified in many scientific disciplines, such as biology, ecology, fishery studies, and social sciences. It appears when the distribution of the collected data is different from that of the target population. For example, in capture-recapture experiments, larger animals are more likely to be captured, so the observed data are a biased sample of the target population. In non-ignorable missing-data problems, the response probability depends on the study variable that is subject to missing; consequently, the distribution of this variable for the completely observed data is different from that of the population, even conditional on covariates. Because of the intrinsic nature of biased sampling problems, valid and effective statistical inference is challenging. Through an extensive literature review, we have seen that existing methods either rely on strong parametric assumptions or suffer from unstable algorithms and efficiency loss in the parameter estimation. This research proposal aims to develop novel methods for biased sampling problems, focusing on inference problems from capture-recapture data and non-ignorable missing data. The first theme will consider the estimation of abundance in a closed population with capture-recapture data. First, we propose to model the capture probabilities via the generalized additive model with shape constraints such as monotonicity for each additive component. Second, we will develop a penalized empirical likelihood (EL) method to find the point estimate and confidence interval (CI) of the abundance; this is an effective method for preventing spuriously large estimates of the abundance. Third, we will design an effective and fast algorithm for calculating the aforementioned point estimate and CI. The second theme will consider non-ignorable missing-data problems. We will model the study variable conditional on the covariates for the completely observed data via a semiparametric Box-Cox transformation model. We will proceed in two steps. In the first step, we will develop a maximum binomial likelihood method to analyze the semiparametric Box-Cox transformation model. In the second step, we will combine this method with the EL method to develop valid estimators for the mean of the study variable and the unknown parameters in the response probability model. The projects in this proposal will develop novel tuning-parameter-free statistical methods to solve important and challenging problems in abundance estimation with capture-recapture data and non-ignorable missing-data problems. These methods will be theoretically solid and implemented in publicly accessible R packages. This research program will in turn benefit theoretical and methodological research into nonparametric likelihood methods and shape-constrained inference. The outcomes will be applicable to a range of scientific problems in Canada arising in health science, economics, wildlife management, and social sciences.