在迴歸分析中，有相關性的資料是很普遍的，如長時間追蹤資料(longitudinal data)、家族間(family)或族群間(cluster)的數據等。藉由將隨機效用加入迴歸模型中，應變數之間的相關性會自然地顯現，我們稱此為隨機效用模型。另一方面共變量也可能無法精準量測而具有測量誤差。當迴歸模型中同時存在測量誤差和隨機效用時，我們稱其為一個混合效用測量誤差模型。分析這樣的模型通常是相當困難的，因為多數適用處理隨機效用的概似函數估計，並不適用於一般的測量誤差模型。因此目前僅有極少數的估計方法是探討混合效用測量誤差模型，而且除了線性混合效用測量誤差模型中的校正分數函數外，其它都只是近似的估計方法，並不具有一致性。 在本文中，我們會說明在線性混合測量誤差模型中，如何推導出條件分數函數。條件分數函數已很廣泛地應用在許多測量誤差模型，除了具有一致性外也常有很好的效率，也由於無需假設共變量的分配而具有穩健性。我們將導證如何在線性混合效用測量誤差模型下得到條件分數函數，並以電腦模擬說明我們的條件分數函數具有以上的優點。

Correlated data like longitudinal observations, family data and cluster data are very common in regression analysis. By introducing the random effect into the regression model, the correlation within subjects emerges naturally. It is also possible that the covariate are subject to measurement error. When measurement error and random effect are both present in a regression analysis, we say that we have a mixed measurement error model-MMeM. The analysis of MMeM is usually difficult for the reason that the likelihood approach which were usually adopted for random effect problems are not applicable for most measurement error problems. Thus, little estimations had been found for MMeM and most of them are based on approximation with an exception on Linear MMeM. In this thesis, we will show how to derive the conditional score in a linear mixed measurement error model. The conditional score is widely used in many measurement error problem, and is robust to the distribution of unobserved covariate. We will show the estimator we proposed for LMMeM also has these advantages.

Contents
1. Introduction 1
2. Models and Corrected scores 2
  2.1 linear mixed  models with measurement error 2
  2.2 Corrected scores function 3
3. Conditional score and the asymptotical distribution 5
  3.1 Estimations that utilize the conditional distribution 8
4. Simulation study 10
5. Conclusion 11
Appendix 19
References 22
Table of contents
table 1 : $n=150 , X_{ij}stackrel{i.i.d.}{sim} N(0,1),,delta_{ij}stackrel{i.i.d.}{sim} N(0,0.5),,b_{i}stackrel{i.i.d.}{sim} N(0,0.5)$ 12
table 2 : $n=150 , X_{ij}stackrel{i.i.d.}{sim} N(0,1),,delta_{ij}stackrel{i.i.d.}{sim} N(0,0.5),,b_{i}stackrel{i.i.d.}{sim} N(0,0.3)$ 12
table 3 : $n=150 , X_{ij}stackrel{i.i.d.}{sim} N(0,1),,delta_{ij}stackrel{i.i.d.}{sim} N(0,0.3),,b_{i}stackrel{i.i.d.}{sim} N(0,0.5)$ 13
table 4 : $n=150 , X_{ij}stackrel{i.i.d.}{sim} N(0,1),,delta_{ij}stackrel{i.i.d.}{sim} N(0,0.3),,b_{i}stackrel{i.i.d.}{sim} N(0,0.3)$ 13
table 5 : $n=150 , X_{ij}stackrel{i.i.d.}{sim} frac{chi^2_1-1}{sqrt{2}},,delta_{ij}stackrel{i.i.d.}{sim} N(0,0.5),,b_{i}stackrel{i.i.d.}{sim} N(0,0.5)$ 14
table 6 : $n=150 , X_{ij}stackrel{i.i.d.}{sim} frac{chi^2_1-1}{sqrt{2}},,delta_{ij}stackrel{i.i.d.}{sim} N(0,0.5),,b_{i}stackrel{i.i.d.}{sim} N(0,0.3)$ 14
table 7 : $n=150 , X_{ij}stackrel{i.i.d.}{sim} frac{chi^2_1-1}{sqrt{2}},,delta_{ij}stackrel{i.i.d.}{sim} N(0,0.3),,b_{i}stackrel{i.i.d.}{sim} N(0,0.5)$ 15
table 8 : $n=150 , X_{ij}stackrel{i.i.d.}{sim} frac{chi^2_1-1}{sqrt{2}},,delta_{ij}stackrel{i.i.d.}{sim} N(0,0.3),,b_{i}stackrel{i.i.d.}{sim} N(0,0.3)$ 15
table 9 : $n=300 , X_{ij}stackrel{i.i.d.}{sim} N(0,1),,delta_{ij}stackrel{i.i.d.}{sim} N(0,0.5),,b_{i}stackrel{i.i.d.}{sim} N(0,sigma^2_b)$ 16
table 10 : $n=300 , X_{ij}stackrel{i.i.d.}{sim} frac{chi^2_1-1}{sqrt{2}},,delta_{ij}stackrel{i.i.d.}{sim} N(0,0.5),,b_{i}stackrel{i.i.d.}{sim} N(0,sigma^2_b)$ 16
table 11 : $X_{ij}stackrel{i.i.d.}{sim} N(0,1),delta_{ij}stackrel{i.i.d.}{sim} N(0,0.3), bstackrel{i.i.d.}{sim} U(-0.5,0.5)sqrt{12ast0.3}$ 17
table 12 : $X_{ij}stackrel{i.i.d.}{sim} N(0,1),delta_{ij}stackrel{i.i.d.}{sim}U(-0.5,0.5)sqrt{12ast0.3}, bstackrel{i.i.d.}{sim} N(0,0.3)$ 17
table 13 : $X_{ij}stackrel{i.i.d.}{sim} N(0,1),delta_{ij}stackrel{i.i.d.}{sim}U(-0.5,0.5)sqrt{12ast0.3} ,bstackrel{i.i.d.}{sim} U(-0.5,0.5)sqrt{12ast0.3}$ 18

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