一般在分析抽樣資料時, 通常是使用傳統的常態近似法。但當資料含有大量零值時, 傳統方法會變得不可靠。利用Kvanli et al (1998) 所提出的最大概度比法處理多零值資料會比傳統方法準確。本文利用 Chen 和 Sitter (1999) 提出將傳統概度函數加權所發展出的擬概度函數, 再結合 Cochran (1977) 的不等機率抽取法來處理具輔助訊息的多零值資料, 其中輔助訊息與多零值資料是有一定相關性。首先將母體依照其相對應的輔助訊息大小排序, 再平分為四群, 然後用不等機率分別從四群中依次抽出10%, 20%, 40%, 30% 的樣本數。為使擬概度函數可具有不偏性,在此是採用相同權數來建構信賴區間。本文是探討在不同的相關係數ρ和非零值比例α , 及不同非零值母體分布下, 比較傳統方法、最大概度比法以及各種最大擬概度比法在95% 名目水準下, 信賴區間的平均值、標準差和涵蓋率的表現。

The many-zero-observation data in survey sampling under complex probability sampling is considered. The confidence interval derived using the traditional normal approximation based on the central limit theorem (CLT) has poor coverage rate even when the sample size is large. The maximum likelihood method (ML) of Kvanli et al (1998) does not work well either. This paper expands the pseudo likelihood method of Chen and Sitter (1999) that integrated unequal probability sampling design (Cochran, 1977). The method utilizes auxiliary information which has a correlation coefficient ρ with the data. A comparison between different methods was conducted under distinct ρ values and nonzero proportion α, and various nonzero distributions. Simulation results indicate that the pseudo likelihood method improves the coverage rate substantially and outperforms the CLT and ML methods.

目錄
1 緒論　1
2 傳統方法與最大概度比法　3
2.1 傳統方法　3
2.2 最大概度比法　3
3 最大擬概度比法　8
3.1 最大擬概度比法　8
3.2 係數 a2n 的估計　14
4 電腦模擬程序　18
5 結論　20
附錄
附錄一 ˆl(α,μ,θ) 為l(α,μ,θ) 的不偏估計式　49
附錄二對數常態和指數擬概度函數推導　50
附錄三  si 和  qij 計算　56
附錄四常態, 伽瑪, 對數常態和指數擬概度比的涵蓋率(%)　61
參考文獻　 66
電腦模擬程式　67
圖表目錄
圖1 各非零值資料的分布圖形　18
表1.1 ~ 1.5 非零值由N(5,1) 分布生成(ρ = 0.0~0.7)　24~28
表2.1 ~ 2.5 非零值由G(7,5/7) 分布生成(ρ = 0.0~0.7) 　 29~33
表3.1 ~ 3.5 非零值由LN(5,0.012) 分布生成(ρ = 0.0~0.7) 34~38
表4.1 ~ 4.5 非零值由E(5) 分布生成(ρ= 0.0~0.7)　39~43
表5.1 ~ 5.5 非零值由LN(5,1) 分布生成(ρ = 0.0~0.7)　44~48
表A1 95% 信賴區間涵蓋率非零值由N(5, 1) 分布生成　 61
表A2 95% 信賴區間涵蓋率非零值由G(7, 5/7) 分布生成　62
表A3 95% 信賴區間涵蓋率非零值由LN(5, 0.012) 分布生成　63
表A4 95% 信賴區間涵蓋率非零值由E(5) 分布生成　 64
表A5 95% 信賴區間涵蓋率非零值由LN(5, 1) 分布生成　65

參考文獻
1. 林慈君(2007) , “不等機率抽樣法求多零值資料的擬概度信賴區間”。淡江大學數學學系碩士論文。
2. 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。
3. 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 Unpublished manuscript.
4. Chen, J. and Sitter, R. (1999), “ A Pseudo Empirical Likelihood Approach to the Effective Use of Auxiliary Information in Complex Surveys ”. Statist. Sinica 9,385-406
5. Cochran, W. G. (1977), Sampling Techniques, 3rd Edition. Wiley, New York.
6. Ijiri, Y. and Leitch, R. W. (1980), “ Stein's Paradox and Audit Sampling ”. Journal of Accounting Research, 18, 91-108.
7. 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.
8. Matsumura, E. M. and Tsui, K. W. (1982), “ Stein-Type Poisson Estimators in Audit Sampling ”. Journal of Accounting Research, 20, 162-170.
9. Press, W. H., Vetterling, W. T., Teukolsky, S. A. and Flannery, B. P. (1993), Numerical Recipes in Fortran, 2nd Edition.
10. Wilks, S. S. (1938), “ The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses ”. Ann. Math. Statist. 9, 60-62.