常被使用的方法，一般我們都假設誤差是來於應變量，但是往往生活當中的測量，誤差是存在於應變量與自變量之中的。因此，當應變量與自變量同時都存在誤差項時，此時我們會利用正交迴歸分析這個統計方法，來分析這兩變量之間所存在的關係。此外，我們測量所拿到的資料，有時候也可能不是確切的數據，而是一筆設限的數據，在此情況下，如果直接用這筆設限的資料來作正交迴歸分析，肯定無法正確分析出兩變量間所具有的關係。在這篇文章我們就是要討論，在設限資料下如何來做正交迴歸的估計，文章中我們提出了一個估計的方法以及兩個判斷的準則，在使用本文所提出的估計方法後且符合所提出準則的情況下，此時所估計出來的正交迴歸線與二變量母體下的正交迴歸線，是沒有明顯差異的。

The method of least squares has been used in general for regression analysis. It is usually assumed that the errors are confined to the dependent variable, but in many cases both dependent and independent variables are typically measured with some stochastic errors. The statistical method of orthogonal regression has been used when both variables under investigation are subject to stochastic errors. Furthermore, the measurements sometimes may not be exact but have been censored. In this situation doing orthogonal regression with censored data directly between the two variables, it may yield an incorrect estimates of the relationship. In this work we discuss the estimation of orthogonal regression under censored data in one variable and then provide a method of estimation and two criteria on when the method is applicable. When the observations satisfy the criteria provided here, there will not be very large differences between the estimated orthogonal regression line and the theoretical orthogonal regression line.

1 Introduction 1
2 Preliminary 2
2.1 Ordinary Regression 2
2.2 Orthogonal Regression 3
3 Estimation Under Censoring Data 5
3.1 An Example with Exact Data and Censored Data 5
3.2 Estimation of Iterative Method 6
4 The Influence of the Censoring Proportion to the Estimation Method Under Censored Data 8
4.1 Criterion One : Ratio of V_1 and V_2 (V_1/V_2) 8
4.1.1 Definitions of V_1 and V_2 8
4.1.2 Formula for V1 and V2 Ratio 9
4.1.3 Simulation Algorithms for V_1/V_2 10
4.1.4 Estimator of V_1/V_2 12
4.2 Criterion Two : Ratio of n_2 and n_1 (n_2/n_1) 16
4.2.1 Definitions of n_1 and n_2 16
4.2.2 Simulation Algorithms for n_2/n_1 17
4.2.3 Ratios n_2/n_1 and V_1/V_2 under Bivariate-T Data 19
5 Non-symmetric Censoring Mechanism 20
5.1 Estimators of Parameters 20
5.2 Simulation Under Non-symmetric Censoring Mechanism 21
6 Conclusion and Discussion 29
Appendix 31
A Some Useful Properties 31
B Simulation Results of V_1/V_2 and n2/n1 34
C Program Procedure 39

[1] Boggs, P. T., Donaldson, J. R. and Schnabel, R. B. (1989). Software for weighted orthogonal distance regression. ACM Transactions on Mathematical Software 15(4), 348-364.
[2] Johnson, R. A. and Wichern, D. W. (2002). Applied Multivariate Statistical Analysis. New Jersey: Prentice-Hall.
[3] Leng, L., Zhang, T., Kleinman, L. and Zhu, W. (2007). Ordinary least square regression, orthogonal regression, geometric mean regression and their applications in aerosol science. Journal of Physics 78(1), 012084.
[4] Montgomery, D. C., Peck, E. A. and Vining, G. G. (2006). Introduction to Linear Regression Analysis. New York: Wiley.
[5] Powell, J. L. (1984). Least absolute deviations estimation for the censored regression model. Journal of Econometrics 25(3), 303-325.
[6] Schaefer, K. C. and Visser, M. L. (2003). Reverse regression and orthogonal regression in employment discrimination analysis. Journal of Forensic Economics 16(3), 283-298.
[7] Wehrens, R., Putter, H. and Buydens, L. M. C. (2000). The bootstrap: a tutorial. Chemometrics and Intelligent Laboratory Systems 54(1), 35-52.
[8] Zwolak, J. W., Boggs, P. T. and Watson, L. T.(2007). Algorithm 869: ODRPACK95: A weighted orthogonal distance regression code with bound constraints. ACM Transactions on Mathematical Software 33(4), 27-38.