本研究主要探討有關抽樣調查的二分類母體比例之估計問題。在抽樣調查當中，有些問題是牽涉到個人隱私或非法的部分行為，對於這些敏感性問題，若是直接調查，經常造成拒答的情形；即使願意回答，亦難確保回答的內容是真實的。為了瞭解敏感性問題的真相，降低回答偏誤以獲取正確的資料，於是有學者提出隨機作答模式。由於隨機作答模式可以視為直接作答模式的一般化方法，且多數研究僅只探討點估計量的部份，本研究採用Wilson (1927)信賴區間建構方法，以數理推導方式求得隨機作答模式的二分類母體比例之一般化點估計量與區間估計量，同時，並推論出各估計量之相關統計性質，此外，亦將進一步以均方誤差與涵蓋機率等評量方式，分別進行點估計量與區間估計量的估計效率比較分析。
    在直接作答模式之下，二個評比的點估計量各有其相對有效區間，而在隨機作答模式之下，由於相對有效區間與設計參數數值有關，因此，相對有效區間必須依據條件成立與否來求得。無論是在直接作答模式或者是在隨機作答模式之下，利用Wilson方法所建構而得之信賴區間表現得均優於Wald信賴區間。本研究並發現二個評比的區間估計量都會有樣本數愈大，愈有可能產生區間上下界限均不合理的問題。

This study considers the problem of estimation for binomial proportions of sensitive attributes in the population of interest. Randomized response models are suggested for protecting the privacy of respondents and reducing the response bias while eliciting information on sensitive attributes. By applying the Wilson (1927) approach for constructing confidence intervals, various probable point estimators and confidence interval estimators are suggested for the common structures of randomized response models. The results also cover to the case of direct response model. Efficiency comparisons are carried out to study the performance of the proposed estimators for both the cases of direct response and randomized response models. In particular, efficiency comparisons are worked out for point estimators comparison and confidence intervals comparison separately. The efficiency aspect of the proposed point estimators is studied with respect to mean square error criterion. To evaluate the performance of confidence intervals, we concentrate on coverage probability. Circumstances under which each proposed estimators is better in use are also identified. In addition, the effects of design parameters will be discussed.
    For the case of direct response model, one of the two competing point estimators is more efficient than the other under certain circumstances. For the case of randomized response model, circumstances under which a point estimator is superior to the other are correlated with design parameters such that it is in need of checking whether the condition holds. For both the cases of direct response and randomized response models, the Wilson approach performs better than Wald confidence interval. It is also found that both the two competing confidence intervals suffer from the undesirable feature that larger sample size results in higher possibility of both the upper and lower limits of the interval outside the parameter space.

目錄
中文摘要…………………………………………………………………I
英文摘要………………………………………………………………II
目錄……………………………………………………………………IV
表目錄…………………………………………………………………VI
圖目錄…………………………………………………………………VII
通用符號一覽表……………………………………………………VIII
第一章	 緒論……………………………………………………………1
  1.1 研究背景與動機…………………………………………………1
  1.2 研究目的…………………………………………………………3
  1.3 論文結構…………………………………………………………4
第二章	 文獻探討………………………………………………………6
  2.1 比例估計…………………………………………………………6
  2.2 隨機作答模式…………………………………………………15
第三章	 隨機作答模式之一般化比例估計……………………………24
  3.1 一般化比例估計………………………………………………24
  3.2 比例估計量之推論……………………………………………28
第四章	 估計效率比較分析……………………………………………31
  4.1 直接作答模式……………………………………………………31
    4.1.1 點估計量比較………………………………………………31
    4.1.2 區間估計量比較……………………………………………34
  4.2 隨機作答模式……………………………………………………39
    4.2.1 點估計量比較………………………………………………39
    4.2.2 區間估計量比較……………………………………………42
第五章	 結論……………………………………………………………50
  5.1 主要研究結果…………………………………………………50
  5.2 未來研究方向…………………………………………………52
參考文獻………………………………………………………………54

表目錄
表4.1	直接作答模式點估計量相對有效區間彙整表………	34
表4.2	直接作答模式信賴區間涵蓋機率彙整表……………	36
表4.3	直接作答模式不合理區間界限相對次數彙整表……	38
表4.4	隨機作答模式信賴區間涵蓋機率彙整表(p=0.7)…44
表4.5	隨機作答模式信賴區間涵蓋機率彙整表(p=0.8)…45
表4.6	隨機作答模式不合理區間界限相對次數彙整表(p=0.7)…47
表4.7	隨機作答模式不合理區間界限相對次數彙整表(p=0.8)…48

圖目錄
圖1.1	本文研究架構圖………………………………………	5
圖2.1	直接作答模式Wald信賴區間示意圖…………………	8
圖2.2	直接作答模式Wald信賴區間涵蓋機率示意圖………	9
圖2.3	直接作答模式Wilson信賴區間示意圖………………	11
圖2.4	直接作答模式Wilson信賴區間涵蓋機率示意圖……	11
圖4.1	直接作答模式點估計量相對效率比較圖……………	33
圖4.2	直接作答模式信賴區間涵蓋機率比較圖……………	35
圖4.3	隨機作答模式點估計量相對效率比較圖……………	41
圖4.4	隨機作答模式信賴區間涵蓋機率比較圖(p=0.7)……43
圖4.5	隨機作答模式信賴區間涵蓋機率比較圖(p=0.8)……43
圖4.6	隨機作答模式Wald信賴區間示意圖…………………	49
圖4.7	隨機作答模式Wilson信賴區間示意圖………………	49

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