Estimating Population Size by Spline Regression Models.

當重複捕取模式含相關輔助變數時，傳統上採用Huggins-Alho 的程序來估計母體總數(Pollock, 2002)。此方法利用一參數化邏輯斯迴歸來描述動物個體之被捕機率。不過，當捕取機率與相關輔助變數間有較複雜的函數關係，這個方法並不適用。Huggins& Hwang (2006); Hwang & Huggins (2006)於是利用非參數核平滑法發展出較具彈性的模式，可以彌補傳統模式的缺點。但是，他們採用的核平滑估計法需要較多的計算時間與成本，尤其是他們建議以經驗偏差帶寬選擇法(empirical bias bandwidth selection,EBBS)來選擇平滑參數，這個步驟更會將計算成本提高很多。本計劃將採用懲罰樣條(penalized spline)迴歸於重複捕取模式上的應用，因為它是非參數核平滑法的另一種選擇。此外，因為它可以被視為一種混合模式，這將使得選擇平滑參數的計算成本降到最低。本計畫將由發展一半參數迴歸模式出發，此模式將包含所有傳統之典型模式(Huggins, 1989)，也將包含Huggins & Hwang(2006)之非參數模式。我們將研究此方法之推估程序並與核平滑法作一比較，最後也會將此法應用到實際例子上。The estimation of the population size for a capture-recapture model with covariates istraditionally based on the Huggins-Alho approach (Pollock, 2002). In their approach, thecapture probabilities are typically related to the covariates via a parametric logistic regressionmodel. However, this model has limitations in modelling complex relationships betweencovariates and the capture probabilities. Huggins & Hwang (2006); Hwang & Huggins (2006)developed more flexible nonparametric models. They used nonparametric and semiparametricmodels, and demonstrated that the kernel smoothing approach can have marked advantages.Nevertheless, their methods usually have a high computational cost. In particular, theyproposed using the empirical bias bandwidth selection (EBBS) procedure to choosesmoothing parameters, such procedure shall increase computational cost very much.This project considers the use of penalized spline regression on capture-recaptureexperiments as it provides an alternative to kernel smoothing in nonparametric regressioncontext. Furthermore, penalized splines can be viewed as a mixed model, which allows usselecting smoothing parameters automatically. We will begin by developing a semiparametricregression model, which can include classical parametric models (Huggins, 1989) and thesemiparametric model of Hwang & Huggins (2006). Then we will study the inferenceprocedures and compare with the results of kernel smoothing. Finally, we will apply theproposed methods to analyze several real data