重複捕取(capture-recapture)實驗是一個常用來估計母體總數的方法，對於不同的重複捕取實驗而言，統計上已經發展出許多相關的分析方法和推論。異質性模型(heterogeneous model)的概念便是其中之一，即是考慮不同個體的異質性，如身高、體重、性別或其他可能影響捕取機率的因子(Pollock(1974))。其中邏輯式迴歸模型可能用於描繪動物被捕取的機率與其自變數之間的關係。理想上，自變數可以在動物被捕取時準確被測量，但實際上，它往往含有隨機誤差。

在過去的文獻中，假設測量誤差的變異數不論捕取次數多寡皆是一樣的，似乎蘊含著每隻動物只會被測量一次自變數，在常理上較少使用。由於並非所有觀測對象只會被捕取一次，有些會被捕取一次以上，被捕取多次的觀測值平均值當然也會比較精準，因此我們應將它們全部使用在我們的估計式中，在本文我們將使用RRC的概念利用觀測值平均值進行迴歸係數和母體總數的估計。最後以模擬方法考量自變數與隨機誤差在不同分配之下估計方法的表現，並計算其平均估計值、估計值的樣本標準差、平均估計標準差、樣本均方根差(RMSE)與95%信賴區間涵蓋率(CP)，並對結果加以討論。

Capture-recapture is a commonly used method in estimating the population size.Ideally, covariate can be measured accurately for the statistical analysis.However it is usually measured with measurement error in practice.Some existent method had been developed to accommodate the measurement error,but these methods assume the variance of measurement error is homogeneous.This assumption implies that for each captured animal,the investigator measured its covariate only once though it may be captured more than one time.

It is quite possible in practice that each animal's covariate is measured everytime whenever it is captured, and the average of the observed value is more accurate than a single observed value.Therefore we constructed an estimating method using the average value based on the concept of refined regression calibration.It turns out that our method is also suitable when the variance of measurement error is heterogeneous.Simulation results are conducted for different kind of variance of measurement error and three kind of distribution of covariates.

1.緒論.....1
2.在自變數有測量誤差下的母體估計方法簡述.....3
 2.1符號簡介.....3
 2.2常用的近似與誤差模型.....4
 2.3Naive簡易方法.....5
 2.4RC(Regression Calibration)簡述.....5
 2.5RRC(Refined Regression Calibration)簡述......5
3.有重複測量下的RRC近似方法.....8
4.模擬研究.....12
 4.1模擬條件.....12
 4.2模擬結果.....13
5.結論與討論..... 15
參考文獻.....16
附錄.....17
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Carroll, R. J., Ruppert, D., and Stefanski, L. A. (2003).Measurement Errors in Nonlinear Models, Chapman & Hall, London.

Huggins, R. M. (1989). On the statistical analysis of capture experiments. Biometrika 76, 133-140.

Hwang, W. H. and Huang, S. Y. H. (2003). Estimation in capture-recapture models when covariates are subject to measurement errors.Biometrics 59, 1115-1124.

Hwang, W. H., Huang, S. Y. H. and Wang C. Y. (2007). Effects of measurement error and conditional score estimation in capture-recapture models.  Satistica Sinica 17, 301-316.

Johnson, N. L., Lotz, S., and Balakrishnan, N. (1995). Continuous Univariate Distributions,Volume 2:Second Edition, Wiley, New York.

Liang, K. Y. and Liu, X. H. (1991). Estimating equations in generalized linear models with measurement error.In Estimating Functions, ed. V.P. Godambe, 47-63, Oxford: Clarendom Press, New York.

Pollock, K. H. (1974). The Assumption of Equal Catchability of Animal in Tag-Recapture Expertiments. Cornell University,New York. Ph. D. Thesis.

Pollock, K. H. (2002). The use of auxiliary variables in capture-recapture modeling: an overview. Journal of Applied Statistics.29, 85-102.

Stefanski, L. A. and Carroll R. J. (1987). Conditional scores and Optimal scores for generalized linear measurement-error models.Biometrika 74, 703-716.

籃揚凱(2006)具有異質性測量誤差的重複捕取實驗之估計，中華民國碩士論文