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系統識別號 U0026-2407201412060400
論文名稱(中文) 調適型集群抽樣下比例估計式的統計推論
論文名稱(英文) Statistical Inference of Ratio Estimation under Adaptive Cluster Sampling
校院名稱 成功大學
系所名稱(中) 統計學系
系所名稱(英) Department of Statistics
學年度 102
學期 2
出版年 103
研究生(中文) 林楓敏
研究生(英文) Feng-Min Lin
學號 R28941013
學位類別 博士
語文別 英文
論文頁數 72頁
口試委員 口試委員-呂金河
口試委員-余清祥
口試委員-溫敏杰
口試委員-馬瀰嘉
指導教授-趙昌泰
中文關鍵字 調適型集群抽樣  最小充份統計量  充份統計量  比例估計  Pseudo-empirical概似方法  信賴區間 
英文關鍵字 Adaptive cluster sampling  Minimal sufficient statistic  Sufficient statistic  Ratio estimator  Pseudo-empirical likelihood method  Confidence interval 
學科別分類
中文摘要 當我們有興趣研究的母體是稀少並且具有群聚現象時,調適型集群抽樣不僅可以提供較傳統抽樣方法更為有效之估計,亦同時提供較大之樣本產量。針對不同的起始抽樣設計,過去研究中已經發展出各種不同之調適型集群抽樣方法,並且也已被廣泛的應用於各式各樣的抽樣調查領域,例如:生態統計、環境統計、社會學及流行病學等等。調適型集群抽樣中最初的不偏估計並不是最小充分統計量的函數,同時樣本中有部分的資料並沒有被使用於這些估計中。過去一些研究人員為了可以進一步改進原先不偏估計以求得更好的推論結果,使用Rao-Blackwellization法推導出利用最小充份統計量所建構之不偏估計,亦有利用其他充份統計量所建構出的不偏估計式。
然而在許多抽樣調查研究中,研究者往往不僅可以收集到所感興趣的主要變數,額外的輔助訊息也常伴隨著被觀察。運用主要變數與輔助變數的相關性可以提升估計精確度,提供更有效率且較精確的推論。於調適型集群抽樣中,首先利用不偏估計所得之比例估計被提出,也提供了較好之估計結果,而那些比例估計仍舊浪費了部分的資料。因此又有利用單變量之Rao-Blackwellized不偏估計所建構之比例估計被提出,而此一改進型比例估計確實也被證實比原先比例估計表現更佳,但是其變異數及建構完整推論所需要之變異數估計則尚未被提出。
本文將提出上述之改進型比例估計的變異數及變異數估計,同時以蒙地卡羅模擬法探討變異數估計的表現。另外,以該變異數估計與有限母體中央極限定理建構信賴區間。因為比例估計中之變異數估計常有隨著輔助變數之母體平均估計而有高估或低估的現象,此類似現象亦存在於調適型集群抽樣,因此本文中亦提出修正之變異數估計所建構的信賴區間,模擬結果顯示修正之變異數估計所建構的信賴區間可以得到較佳的表現。
另一方面,適合調適型集群抽樣的母體往往有嚴重右偏傾向,且在過去研究中即發現在這種情況下,由有限母體中央極限定理建構之信賴區間可能有涵蓋率偏低的問題。因此,本文進一步探討利用無母數方法來建構信賴區間。於本文第二部份,我們考慮輔助訊息,提出兩種pseudo-empirical概似比例信賴區間,並且將他們與改進型比例估計建構的常態漸進信賴區間做比較,模擬結果呈現pseudo-empirical概似比例建構之信賴區間表現略佳。
英文摘要 Adaptive cluster sampling is able to provide more efficient estimators of the population quantity of interest together with more abundant sampling yields compared to the conventional sampling designs when the population is a rare and clustered one. Various adaptive sampling designs with respect to different initial conventional designs have been developed in the past, and they have been applied in different disciplines, such as environmental research, ecological research, sociology, and epidemiology studies. Originally, the unbiased estimators in adaptive cluster sampling are not the functions of the minimal sufficient statistic, and certain information obtained from the sample is not utilized in the estimators. For better estimation results, different Rao-Blackwellized unbiased estimators based on the minimal sufficient statistic and certain sufficient statistic also have been proposed in the past, and successfully provided better estimation results.

Often certain auxiliary variables would also be available in a sampling survey situation, and one would like to utilize this auxiliary information into the estimation so that he can take advantage of the correlation between the primary and auxiliary variables. Ratio estimators which
make use of the original unbiased estimators have been proposed, and they are able to provide better estimation results when the primary and auxiliary variables are related in a certain degree. In addition, improved ratio estimators based on the univariate Rao-Blackwellized estimators are also proposed in past researches, and they can effectively outperform the original ratio estimators.
Nevertheless, the variances and variance estimators of the improved ratio estimators are still unavailable, hence it is of both practical and theoretical interest to investigate the related issue in order to establish a complete inference.

In this article, the variances and the associated variance estimators of these improved ratio estimators are proposed for a thorough framework of statistical inference under adaptive cluster sampling. Performance of the proposed variance estimators is evaluated in terms of the absolute relative percentage bias and the empirical mean-squared error. As expected, results show that both the absolute relative percentage bias and the empirical mean-squared error decrease as the initial sample size increases for all the variance estimators. To evaluate the confidence intervals based on these variance estimators and the finite population Central Limit Theorem, the coverage rate and the interval width are employed. These confidence intervals suffer similar disadvantage as that of the conventional ratio estimator. Hence, alternative confidence intervals based on a certain type of adjusted variance estimators are constructed and assessed in this article.

Additionally, the population in which adaptive cluster sampling would be an appropriate sampling design is
often highly skewed, hence the CLT-based confidence interval often fails to provide appropriate coverage probability. Hence, it is of interest to further study on the feasibility of non-parametric type of confidence intervals based on the effective usage of auxiliary information. In the second part of this article, we propose two types of pseudo-empirical likelihood ratio confidence intervals based on the usage of auxiliary information and compare the performance of the pseudo-empirical based confidence intervals with the CLT-based confidence intervals based on the ratio estimators using a simulation study. The simulation results show that the confidence intervals obtained from both types of pseudo-empirical likelihood methods perform slightly better.
論文目次 1 Introduction 1
2 ACS and the associated estimators 6
2.1 Sampling methodology of ACS . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Unbiased estimators in ACS . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Original unbiased estimators . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Rao-Blackwellized estimators . . . . . . . . . . . . . . . . . . . 10
2.3 Ratio estimators in ACS . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Original ratio estimators . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Improved ratio estimators . . . . . . . . . . . . . . . . . . . . . 17
3 Variances and variance estimators of the improved ratio estimators 20
3.1 Horvitz-Thompson type . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Hansen-Hurwitz type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Confidence interval of the improved ratio estimators 33
4.1 Simulated population . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Blue-winged teal data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Pseudo-empirical likelihood ratio confidence intervals incorporating auxiliary information
5.1 Pseudo-empirical likelihood ratio confidence intervals . . . . . . . . . . 44
5.1.1 Pseudo-empirical likelihood for conventional sampling . . . . . . 45
5.1.2 Pseudo-empirical likelihood for ACS . . . . . . . . . . . . . . . 47
5.2 A pseudo-empirical likelihood approach to the effective use of auxiliary information in ACS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2.1 Hansen-Hurwitz type pseudo-empirical likelihood ratio confidence interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.2 Horvitz-Thompson type pseudo-empirical likelihood ratio confidence interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3.1 Simulated populations . . . . . . . . . . . . . . . . . . . . . . . 57
5.3.2 Blue-winged data . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6 Discussion and conclusion 60
Appendix A Derivation of the bound for MSE(ˆμr·ht)
Appendix B Derivation of the variance and variance estimator of ˆμr·ht(s) 65
Appendix C Proof of the asymptotic distribution for the pseudo-empirical log-likelihood ratio function rrhh(θ) 67
Bibliography 69
List of Tables
3.1 An illustrative population . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Joint initial intersection probabilities . . . . . . . . . . . . . . . . . . . 27
4.1 The coverage rates and the interval widths of the 95% confidence intervals associated with the Horvitz-Thompson type ratio estimators for the population listed in Figure 3.1 . . . . . . . . . . . . . . . . . . . . . . 35
4.2 The coverage rates and the interval widths of the 95% confidence intervals associated with the Hansen-Hurwitz type ratio estimators for the population listed in Figure 3.1 . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Blue-winged teal data (Smith et al. 1995) . . . . . . . . . . . . . . . . 39
4.4 The coverage rates and the interval widths of the 95% confidence intervals associated with the Horvitz-Thompson type ratio estimators for the blue-winged teal data . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 The coverage rates and the interval widths of the 95% confidence intervals associated with the Hansen-Hurwitz type ratio estimators for the blue-winged teal data . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1 Results on 95% confidence intervals associated with the Hansen-Hurwitz type ratio and pseudo-empirical likelihood methods for the population listed in Figure 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Results on 95% confidence intervals associated with the Horvitz-Thompson type ratio and pseudo-empirical likelihood methods for the population listed in Figure 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3 Results on 95% confidence intervals associated with the Hansen-Hurwitz type ratio and pseudo-empirical likelihood methods for the blue-winged teal data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4 Results on 95% confidence intervals associated with the Horvitz-Thompson type ratio and pseudo-empirical likelihood methods for the blue-winged teal data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
List of Figures
2.1 An example of a network, a edge unit and a cluster . . . . . . . . . . . 8
3.1 A simulated Poisson cluster population of size N = 400 units containing two parents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Absolute relative percentage bias and empirical MSE . . . . . . . . . . 32
4.1 The scatter plots of dAvar(ˆμr·ht(·)) and gAvar(ˆμr·ht(·)) vs. ˆμx·ht(·) under the simulated population in Fig. 3.1 with initial sample size n = 40. . . . . 36
4.2 The scatter plots of dAvar(ˆμr·hh(·)) and gAvar(ˆμr·hh(·)) vs. ˆμx·hh(·) under the simulated population in Fig. 3.1 with initial sample size n = 40. . . 37
4.3 The average widths and the coverage rates (%) of the 95% confidence intervals associated with dAvar(ˆμr··(·)) and gAvar(ˆμr··(·)) under the population in Figure 3.1 with n0 = 40 . . . . . . . . . . . . . . . . . . . . . 38
4.4 The average widths and coverage rates (%) of the 95% confidence intervals associated with dAvar(ˆμr·i(ms)) and gAvar(ˆμr·i(ms)) under the bluewinged teal data with n0 = 8 . . . . . . . . . . . . . . . . . . . . . . . . 42
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