在生態學中，調查對象的密度或總數是相當重要的參數，而距離抽樣則是方便且常採用
的抽樣方法，本文討論的距離抽樣是直線穿越抽樣法，並且是觀測距離無法精準量測而有測量誤差時的統計分析問題。在直線穿越抽樣法有乘法性測量誤差時，本文將乘法性測量誤差模型作自然對數轉換，使轉換後的模型為相加性測量誤差模型，並比較相加性測量誤差時常見的修正方法：迴歸校正(regression calibration)、泰勒(Taylor)展開式及模擬外推法(simulation extrapolation)等的電腦模擬表現。

In ecology, the density or the total number of residents are very important parameters. To estimate these parameters, the distance sampling is a convenient and often used sampling methods. We discussed this sampling method in line transect experiment when covariates are subject to multiplicative measurement errors. We evaluate the performances of methods of regression calibration, Taylor approximation and simulation extrapolation by simulation
studies.

目錄
1 緒論1
2 線穿越模型中有乘法性測量誤差時之近似估計方法 3
2.1 符號定義 3
2.2 機率密度估計法 5
2.3 乘法測量誤差模型 7
2.4 Naive及其它修正方法 8
3 模擬研究 14
3.1 模擬條件 14
3.2 模擬結果 15
4 結論17
參考文獻18
附錄A 各種修正方法與真實數值的模擬比較18

表目錄
1 偵測函數  = 10,D=2.5,( Z,  Z)=(-0.045,0.3),L=10,N=1500 20
2 偵測函數  = 30,D=2.5,( Z,  Z)=(-0.045,0.3),L=10,N=1500 20
3 偵測函數  = 10,D=2.5,( Z,  Z)=(-0.125,0.5),L=10,N=1500 20
4 偵測函數  = 10,D=2.5,( Z,  Z)=(-0.045,0.3),L=30,N=4500 21
5 偵測函數  = 30,D=2.5,( Z,  Z)=(-0.0125,0.5),L=10,N=1500 21
6 偵測函數  = 30,D=2.5,( Z,  Z)=(-0.045,0.3),L=30,N=4500 21
7 偵測函數  = 10,D=2.5,( Z,  Z)=(-0.125,0.5),L=30,N=4500 22
8 偵測函數  = 30,D=2.5,( Z,  Z)=(-0.125,0.5),L=30,N=4500 22
9 偵測函數  = 10,D=1.5,( Z,  Z)=(-0.045,0.3),L=10,N=900 23
10 偵測函數  = 30,D=1.5,( Z,  Z)=(-0.045,0.3),L=10,N=900 23
11 偵測函數  = 10,D=1.5,( Z,  Z)=(-0.125,0.5),L=10,N=900 23
12 偵測函數  = 10,D=1.5,( Z,  Z)=(-0.045,0.3),L=30,N=2700 24
13 偵測函數  = 30,D=1.5,( Z,  Z)=(-0.0125,0.5),L=10,N=900 24
14 偵測函數  = 30,D=1.5,( Z,  Z)=(-0.045,0.3),L=30,N=2700 24
15 偵測函數  = 10,D=1.5,( Z,  Z)=(-0.125,0.5),L=30,N=2700 25
16 偵測函數  = 30,D=1.5,( Z,  Z)=(-0.125,0.5),L=30,N=2700 25

Carroll, R. J., Ruppert, D., Stefanski, L. A., and Crainiceanu, C.M. (2006). Measurement Errors in Nonlinear Models: A Modern Perspective. Second Edition. Chapman & Hall, London.

Marques, T. A. (2004). Predicting and Correcting Bias Caused
by Measurement Error in Line Transect Sampling Using Multiplicative Error Models. Biometrics, 60, 757-763.

Fan, K. S. (2009). Measurement Errors and Imperfect Detection Rates on theTransect Line in Independent Observer Line Transect Surveys. Journal of Data Science 7, 331-347.

Chen, S. X. (1998). Measurement Errors in Line Transect Surveys. Biometrics 54, 899-908.

Chen, S. X. (2001). Measurement Errors in Line Transect SurveysWhere Detectability Varies with Distance and Size. Biometrica 57, 732-742.18