在迴歸分析時, 我們常會碰到資料無法精準地測量或無法直接觀測的情況。 而若自變數有測量誤差, 此誤差對於估計迴歸係數會造成偏差。 過去的文獻對於重複捕取實驗 (capture-recapture experiment) 的異質性模型 (heterogeneity model) 中自變數含有測量誤差時, 造成估計的偏差的情況, 並未加以深入探討。 Hwang 與 Huang (2003 與 2005) 首先分別以 RRC (refined regression calibration) 方法與 CS (conditional score) 方法來修正迴歸參數估計量的偏差及提出估計母體總數的方法。

本文將探討在封閉性母體 (closed population) 假設下離散型重複捕取實驗中, 若自變數含有小測量誤差 (small measurement error) 時, 如何以泰勒展式近似原始的估計方程式與母體估計量來進行分析, 進而提供一個方便、 有效率且無需許多假設的估計方法。 最後我們以模擬方法考量自變數與隨機誤差在不同分配之下各種估計方法的表現, 並計算不同估計方法的平均估計值、 相對偏差 (RB) 與估計值的樣本標準差、 平均估計標準差、 樣本均方根誤差 (RMSE) 與 95\% 信賴區間涵蓋率 (CP), 最後並對所得的結果加以討論。

In a regression analysis, it may happen that variables are not measured precisely. When the covariates are measured with measurement errors, the estimation in regression parameters will be biased in usual. However, for the capture-recapture experiments, the investigations into the effect of measurement errors have been very limited.

Besides the approaches of regression calibration and conditional score that proposed by Hwang and Huang (2003, 2005), the present paper disscusses an approximate estimation through Taylor expansion for the discrete time capture-recapture experiment. As a result, we find that the small measurement error approximation is a convenient and efficient way for estimaing both in regression parameters and in the population size. Simulation results are also provided for different distributions of measurement error and covariate.

1 緒論 1
2 文獻回顧 3
  2.1 模型假設 3
  2.2 符號簡介 3
  2.3 條例概似模型與估計方法 4
  2.4 在自變數有測量誤差下的母體總數估計方法 6
      2.4.1 Naive 方法 6
      2.4.2 RRC (Refined Regression Calibration) 方法 6
      2.4.3 CS (Conditional Score) 方法 8
3 在小測量誤差下的近似估計方法 (SEA-Small Error Approximation) 10
4 模擬研究 16
  4.1 模擬條件 16
  4.2 模擬結果 18
5 結論與討論 20
參考文獻 21
附錄     23

Carroll, R. J. and Stefandski, L. A. (1995). Approximate quasi-likelihood estimation in Models with surrogate 
predictors. Journal of the Ameriance Statistical Assocation 85, 652-663.

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

Carroll, R. J., Spiegelman, C., Lan, K. K., Bailey K. T., and Abbott R. D. (1984). In errors-in-variables for binary 
regression models. Biometrika 71, 19-26.

Fuller, W. A. (1987). Measurement Error Models. John Wiley & Sons, New York.

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

Huggins R. M., and Yip P. S. F. (1999). Estimation of the size of an open population from capture-recapture data using weighted martingale methods. Biometrics 55. 387-395.

Hwang W. H. and Chao, A. (2002). Continuous-time capture recapture models with covariates. Statistica Sinica 12, 1115-1131.

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)

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. (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.