本文主要討論在重複捕取實驗 (capture-recapture experiment) 下,母體總數 (population size) 的估計問題。文中分別針對封閉型母體 (closed population) 與開放型母體 (open population),提出參數估計方法及估計性質。
在封閉型母體中,本文提出以馬可夫鏈模式 (Markov chain model) 來處理影響捕取機率的三個變異來源:行為反應 (behavior response)、時間效果 (time effect) 與個體差異 (individual heterogeneity)。對於行為反應的處理,傳統行為反應模式認為,捕取機率與之前是否被捕有關。有別於傳統行為反應模式,本文提出以馬可夫相關性 (Markovian dependence) 來架構樣本間的相關,認為已知過去捕取歷史下,捕取機率只與前次是否被捕有關。當時間效果也存在時,此時捕取機率為時間的函數,

可利用非均勻 (nonhomogeneous) 馬可夫鏈同時處理時間效果與行為反應。對於個體差異現象,文中是利用具有隨機效應的捕取機率來解釋,或是透過觀察到的共變量 (covariate) 訊息來估計各動物的捕取機率,分別提出混合效果模式和共變量模式來分析。

在各模式下,針對主要關心的母體總數以及其他相關參數,提出最大概似估計,並討論其估計性質。文中以兩組資料作為實例,Nichols et al. (1984) 分析的北美田鼠資料與 Otis et al. (1978) 的老鼠資料,說明估計方法的實際應用。同時,為了瞭解所提出估計式的表現,文中採用模擬分析加以探討。透過模擬結果發現,在滿足馬可夫相關性的情形下,馬可夫鏈模式確實可以輔助傳統行為模式,降低估計偏差 (bias) 與均方根誤差(root mean squared error),並且得到滿意的涵蓋率 (coverage)。

在開放型母體中,本文主要利用核平滑法 (kernel smoothing approach) 來處理長期性重複捕取實驗中,捕取樣本 (sampling occasions) 很多的問題。對於母體不再滿足封閉性的假設,文中分別提出局部常數模式 (local constant model) 和局部多項式模式 (local polynomial model),逐步放鬆封閉性的要求。對於存在動物個體間的異質現象,模式中將動物的捕取機率視為一組來自未知分配的隨機樣本,此分配由前兩階動差 (moment) 決定,透過隨機性來模式化個體差異的影響,並透過結合樣本涵蓋 (sample coverage) 與估計方程式(estimating equation) 的方法,估計母體總數及其他相關參數,同時推廣了 Huggins 與 Yip (1999) 處理時間效果的估計方法和Chao et al. (2001) 處理封閉母體的估計方法。

對於開放母體總數的變異數估計,除了提出大樣本近似估計外,也介紹如何利用重抽法 (bootstrap method) 的概念來估計變異數,進而得到信賴區間估計。對於核平滑估計中常遇到的最佳帶寬 (optimal bandwidth) 的選取問題,也可利用重抽法來解決,這樣的方法概念上很簡單,電腦計算時間也合理。本部分以香港米埔鳥類保育區的鷦鶯資料,和澳洲南方費雪島上的短尾海鷗資料作為實例,說明所提出的估計方法的實際應用。文中同時藉由一些模擬研究來瞭解核平滑估計式的表現。由模擬分析中發現,在個體異質性真的存在時,所提出的估計方法的確可以改善只考慮時間效果的估計方法,得到較小的均方根誤差。

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