Novel statistical methods for biased sampling problems

有偏抽样在生物学、生态学、渔业研究和社会科学等许多科学学科中都已被发现。当收集的数据的分布与目标人口的分布不同时，就会出现这种情况。例如，在捕获-重新捕获实验中，更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见的样本。在不可忽略的缺失数据问题中，响应概率取决于可能缺失的研究变量；因此，对于完全观察到的数据，该变量的分布与总体的分布是不同的，甚至是以协变量为条件的。由于有偏抽样问题的内在本质，有效和有效的统计推断是具有挑战性的。通过广泛的文献回顾，我们已经看到，现有的方法要么依赖于强参数假设，要么算法不稳定，参数估计效率低下。这项研究的目的是开发新的方法来解决有偏抽样问题，重点是从捕获-重新捕获数据和不可忽略的丢失数据进行推断的问题。第一个主题将考虑利用捕获-重新捕获数据估计封闭种群的丰度。首先，我们建议通过具有形状约束的广义加法模型来建模捕获概率，例如每个加法分量的单调性。其次，我们将开发一种惩罚经验似然(EL)方法来寻找丰度的点估计和可信区间(CI)；这是防止错误地估计丰度的有效方法。第三，我们将设计一种有效而快速的算法来计算前述点估计和CI。第二个主题将考虑不可忽视的数据缺失问题。我们将通过半参数Box-Cox变换模型对研究变量进行建模，条件是完全观测数据的协变量。我们将分两步进行。在第一步中，我们将发展一个最大二项似然方法来分析半参数Box-Cox变换模型。在第二步中，我们将把这种方法与EL方法结合起来，为响应概率模型中的学习变量和未知参数的平均值开发有效的估计。该方案中的项目将开发新的无参数调整的统计方法，以解决捕获-重新捕获数据丰度估计中的重要和具有挑战性的问题，以及不可忽略的丢失数据问题。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。这项研究计划反过来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。这些结果将适用于加拿大在健康科学、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.