我們通常都會使用傳統的常態近似方法計算信賴區間，但是當處理包含大量零值類型的資料時，常態近似法結果會變得相當不準確。 Kvanli、Shen和Deng（1998）提出最大概度比法來處理這種資料，所建立出來的信賴區間比傳統常態近似方法更加準確。本文利
用 Chen和Sitter（1999）將概度函數加權所發展出的擬概度函數方法來分析含有大量零值的資料，依輔助訊息大小排序後分群，使用 Cochran（1977）提出的不同機率抽取樣本的模式，再以不同權數和相同權數建立信賴區間。並探討在各種相關係數ρ與非零值比例α下，信賴區間上下界的平均值和涵蓋率的表現。

In survey sampling, traditional normal approximation is commonly used to construct confidence intervals of the finite population mean. However, when the finite population contains a large proportion of zeroes, the normal approximation may have very poor coverage rate even when the sample size is large. Kvanli, Shen and Deng (1998) propose a parametric likelihood approach to construct a confidence interval and demonstrate that the likelihood ratio based confidence interval has more precise coverage rate. Chen and Sitter (2002) propose a pseudo likelihood function to overcome the difficulties of lacking of exact likelihood. The approach can be used in the present problem. We first sort the corresponding auxiliary information from the smallest to the largest and divide them equally into several groups, then draw a sample according to an unequal probability sampling design (see Cochran 1977). We develop pseudo likelihood ratio intervals using two different weights and discuss their performance with respect to correlation coefficient ρ and nonzero proportion α, and also analyze their lower and upper average bounds and coverage rates.

1  緒論...................................................4
2  概度比與擬概度比方法...................................6
   2.1 最大概度比法.......................................6
       2.1.1 最大常態概度比法.............................6
       2.1.2 最大伽瑪概度比法.............................8
   2.2 最大常態擬概度比法.................................9
   2.3 最大伽瑪擬概度比法................................11
   2.4 估計a^2_n.........................................13
   2.5 權數..............................................15
3  電腦模擬程序..........................................17
4  結論..................................................19
附錄.....................................................31
參考文獻.................................................35
電腦模擬程式.............................................36
表目錄
表4.1....................................................23
4.1.1 95% 信賴區間上下界沒涵蓋的百分比
      非零值由N(5,1)分配生成(rho=0.00)...................23
4.1.2 95% 信賴區間上下界平均值 (標準差)(rho=0.00)........23
表 4.2...................................................24
4.2.1 95% 信賴區間上下界沒涵蓋的百分比
      非零值由N(5,1)分配生成(rho=0.10)...................24
4.2.2 95% 信賴區間上下界平均值 (標準差)(rho=0.00)........24
表 4.3...................................................25
4.3.1 95% 信賴區間上下界沒涵蓋的百分比
      非零值由N(5,1)分配生成(rho=0.30)...................25
4.3.2 95% 信賴區間上下界平均值 (標準差)(rho=0.30)........25
表 4.4...................................................26
4.4.1 95% 信賴區間上下界沒涵蓋的百分比
      非零值由N(5,1)分配生成(rho=0.50)...................26
4.4.2 95% 信賴區間上下界平均值 (標準差)(rho=0.50)........26
表 4.5...................................................27
4.5.1 95% 信賴區間上下界沒涵蓋的百分比
      非零值由GAMMA(25,1/5)分配生成(rho=0.00)............27
4.5.2 95% 信賴區間上下界平均值 (標準差)(rho=0.00)........27
表 4.6...................................................28
4.6.1 95% 信賴區間上下界沒涵蓋的百分比
      非零值由GAMMA(25,1/5)分配生成(rho=0.10)............28
4.6.2 95% 信賴區間上下界平均值 (標準差)(rho=0.10)........28
表 4.7...................................................29
4.7.1 95% 信賴區間上下界沒涵蓋的百分比
      非零值由GAMMA(25,1/5)分配生成(rho=0.30)............29
4.7.2 95% 信賴區間上下界平均值 (標準差)(rho=0.30)........29
表 4.8...................................................30
4.8.1 95% 信賴區間上下界沒涵蓋的百分比
      非零值由GAMMA(25,1/5)分配生成(rho=0.50)............30
4.8.2 95% 信賴區間上下界平均值 (標準差)(rho=0.50)........30

1.葉美銀 (2004) ,"擬概度比法在不同機率抽樣的應用"。淡江大學數學學系碩士論文。

2.Chen, H. ,Chen, J. and Chen, S-Y. (2006),“Confidence Intervals for the Mean  of  a  Population  Containing  Many  Zero  Values  under  Unequal Probability Sampling”.Unpublished manuscript.

3.Chen,  J.  ,Chen,  S.Y.  and  Rao,  J.N.K.(2003),“  Empirical  Likelihood Confidence intervals for The Mean of Population Containing Many Zero Values ”.  The Canadian Journal of Statistics,31(1),53-68.

4.Chen,  J.  and  Qin,  J.  (1993),  “  Empirical  Likelihood  Estimation  for Finite Populations and the Effective Usage of Auxiliary Information ”. Biometrika, 80, 107-116.

5.Chen, J. and Sitter, R. (1999), “ A Pseudo Empirical Likelihood Ap- proach to the Effective Use of Auxiliary Information in Complex Sur- veys ”.  Statist.  Sinica 9, 385-406

6.Cochran, W. G. (1977), Sampling Techniques, 3rd Edition. Wiley, New York.

7.Ijiri, Y. and Leitch, R. W. (1980), “ Stein’s Paradox and Audit Sam- pling ”.  Journal of Accounting Research, 18, 91-108.

8.Kvanli,  A. H. ,  Shen,  Y. K. and Deng,  L. Y. (1998),  “ Construction of Confidence Intervals for the Mean of a Population Containing Mary Zero Values ”. Journal of Business & Economic Statistics, 16, 362-368.

9.Matsumura, E. M. and Tsui, K. W. (1982), “ Stein-Type Poisson Es- timators  in  Audit  Sampling  ”.   Journal  of  Accounting  Research,  20, 162-170.