在本文中,我們探討離散型重複捕取實驗的自變數有異質性測量誤差的估計方法。在重複捕取實驗中至少有兩種情況應考慮異質性測量誤差。一個是測量誤差大小可能與真實自變數大小有關;另一個是個體被重複測量的次數會因捕取次數不同而造成測量誤差大小不同。
因此異質性誤差的假設,相較於大部分文獻所採用的同質性誤差的假設,顯得實際得多。我們使用無母數迴歸來估計測量誤差的變異數,並在小測量誤差的假設下,以泰勒展開式近似原始的分數函數,以及母體總數的估計量,除此之外不需其他太多假設。當測量誤差的變異數趨近於0時,我們的估計方法會趨近於使用原始條件概似函數的估計方法,由於我們的修正方法相當有彈性,可以很容易地修改並應用於其他的重複捕取模型。最後,我們利用電腦模擬評估我們所提議的估計方法。

In this paper, we consider the estimation of discrete time capture-recapture model when covariate is subject to heterogeneous measurement error. There are two reasons for considering heterogeneous error in the capture-recapture experiments. One reason is for the possibility that the magnitude of the measurement error may depend on the true covariate, and the other reason is for the numbers of replicate measurements in each individual can be very different. Thus, a heterogeneous error assumption is more practical than the homogeneous error assumption which was adopted by most literatures. We used a nonparametric regression to estimate the variance of measurement error, 
and used Taylor polynomial to estimate the original score function and population size estimator under small measurement error assumption. There are not much assumptions required to implement the proposed method. 
Our estimation will reduce to the conditional likelihood approaches when the error's variance approaches 0, and is easy to modify at least in theory to accommodate many kinds of heterogeneous error or capture-recapture models.Some simulation studies were conducted to assess the performances of the proposed estimation.

1.緒論(1)
2.離散型重複捕取模型及現有分析方法(3)
  2.1符號簡介 (3)
  2.2模型假設與估計方法 (4)
  2.3有測量誤差時現有的估計方法(5)
     2.3.1 Naive 方法 (5)
     2.3.2 RRC (Refined Regression Calibration) 方法(6)
     2.3.3 CS (Conditional Score) 方法 (7)
3.在小測量誤差下的近似估計方法 (8)
  3.1 異質性測量誤差變異數的估計 (8)
  3.2 小測量誤差下估計函數的近似 (9)
4.模擬研究 (14)
  4.1 模擬條件(14)
  4.2 模擬結果(15)     
5.結論與討論 (17)
參考文獻 (18)
附錄 (19)
  附錄A (19)
  附錄B (22)
      表1 (22)
      表2 (23)
      表3 (24)
      表4 (25)
      表5 (26)
      表6 (27)
      表7 (28)
      表8 (29)
      表9 (30)
      表10 (31)

Carroll, R. J., Ruppert, D., and Stefanski, L. A. (1995). Measurement Errors in Nonlinear Models, Chapman Hall, London.

Cheng, C. L. and Van Ness, J. W. (1998). Statistical Regression with Meansurement Error, Oxford University Press, Arnold, London.

Hardle,Wolfgang (1991).Applied Nonparametric Regression, Cambridge University Press.

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. (2005). Effects of measurement error and conditional score estimation
in capture-recapture models. (Submitted)

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

Stefanski, L. A. (1985). The effects of measurement error on parameter estimation. Biometrika 72, 583-592.

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

陳飛穎 (2005) 重複捕取實驗中自變數有測量誤差時的近似估計方法,中華民國碩士論文。