Novel statistical methods for biased sampling problems

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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.