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研究生： 高菲菲
Kao, Fei-Fei
論文名稱： 系統抽樣修正方法的研究
A Study on the Modification of Systematic Sampling
指導教授： 呂金河
Leu, Ching-Ho
學位類別： 博士
Doctor
系所名稱： 管理學院 - 統計學系
Department of Statistics
論文出版年： 2006
畢業學年度： 94
語文別： 英文
論文頁數： 118
中文關鍵詞： 系統抽樣法 、兩階段抽樣法 、馬可夫連抽樣法 、Horvitz-Thompson 估計量 、修正中央環狀系統抽樣法 、修正中央環狀系統抽樣法 、餘數馬可夫抽樣法 、兩階段馬可夫抽樣法 、空間餘數抽樣法
外文關鍵詞： Markov chain designs, two-stage sampling, systematic sampling, modified balanced circular systematic sampling, Horvitz-Thompson estimator, modified centered circular systematic sampling, remainder Markov sampling, spatial remainder sampling., two-stage Markov sampling
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    • 當母體大小 $N$ 無法被樣本大小 $n$ 整除時，若執行系統抽樣法，則抽樣間距 $k$ 不易決定，且樣本均值為母體均值之有偏估計量。為解決此類問題並提升估計的有效性，本研究提出一些系統抽樣的改良方法。 我們提出修正均衡環狀系統抽樣法(modified balanced circular systematic sampling)，修正中央環狀系統抽樣法 (modified centered circular systematic
    sampling)， 餘數馬可夫抽樣法 (remainder Markov sampling)及兩階段馬可夫抽樣法 (two-stage Markov sampling)等新的抽樣法。更進一步地，探討二維空間抽樣，當母體的行與列大小 $N$, $M$ 無法被樣本大小 $n$, $m$ 整除時，建議使用 空間餘數抽樣法(spatial
    remainder sampling)。我們對提出的新方法分別找出第一階和第二階包含機率，並使用Horvitz-Thompson 估計量為母體均值的不偏估計量，進而推導出估計量之變異數及討論在不同超母體 (super-population)之下所提修正抽樣法的有效性。

    When the population size $N$ is not a multiple of sample size $n$,the sample interval $k$ is not easy to determine and the usual systematic sampling design results in a variable sample size, in this case the sampling mean will be a biased estimator for the population mean. To obviate these difficulties and in some cases to
    increase the efficiency of the estimation of population mean, a number of modifications of systematic sampling will be proposed in this thesis. New sampling methods called modified balanced circular systematic sampling, modified centered circular systematic sampling, remainder Markov sampling and two-stage Markov sampling are
    introduced. Furthermore, consider a spatial population where the units are arranged in a rectangular array of $M$ rows and $N$ columns from which a sample size $mn$ is to be drawn. We propose the spatial remainder sampling procedure when the population size $N$ is not a multiple of sample size $n$, and $M$ is not a multiple of $m$. First and second-order inclusion probabilities are derived for new sampling designs, yielding the Horvitz-Thompson estimator and its variance. Moreover, we also compare the efficiencies of the proposed sampling designs for various of super-populations.

    1. Introduction 1 1.1 The Basic Procedure 1 1.2 Problems in Systematic Sampling 2 1.2.1 Variance Estimation 3 1.2.2 The Case N is unequal to nk 5 1.2.3 Trends and Periodic Components 6 1.3 The Proposed Sampling Procedures 8 2. Modified Circular Systematic Sampling 9 2.1 Introduction 9 2.2 Modified Circular Systematic Sampling Procedures 10 2.2.1 Modified Balanced Circular Systematic Sampling 10 2.2.2 Modified Centered Circular Systematic Sampling 11 2.3 Estimator for the Population Mean 11 2.3.1 Population with Linear Trend 11 2.3.2 Population with Parabolic Trend 12 2.4 Super-population Model 13 2.5 Concluding Remarks 19 3. Remainder Markov Sampling with One Unit Per Stratum 20 3.1 Introduction 20 3.2 Remainder Markov Sampling Procedures 21 3.3 Horvitz-Thompson Estimation under Remainder Markov Sampling 27 3.4 Efficiency of Remainder Markov Sampling 28 3.4.1 Population in Random Order 28 3.4.2 Population with Linear Trend 29 3.4.3 Population with Exponentail Trend 37 3.4.4 Auto-correlated Population 45 3.5 Example 55 3.6 Concluding Remarks 57 4. Two-stage Markov Sampling 59 4.1 Introduction 59 4.2 Two-stage Markov Sampling Procedures 60 4.3 Horvitz-Thompson Estimation under Two-stage Markov Sampling 63 4.4 Efficiency of Remainder Markov Sampling 64 4.4.1 Population in Random Order 64 4.4.2 Population with Linear Trend 65 4.4.3 Auto-correlated Population 69 4.5 Concluding Remarks 73 5. Spatial Remainder Sampling 75 5.1 Introduction 75 5.2 Spatial Remainder Sampling Procedures 76 5.2.1 Spatial Remainder Stratified Sampling Procedures 77 5.2.2 Spatial Remainder Linear Systematic Sampling Procedures 79 5.3 Horvitz-Thompson Estimation under Remainder Spatial Sampling 81 5.4 Evaluation on the Finite Population Variances 85 5.5 Efficiency of Spatial Sampling 101 5.5.1 Population in Random Order 102 5.5.2 Population with Linear Trend 103 5.5.3 Population with Spatially Correlated 106 5.6 Example 107 5.7 Concluding Remarks 108 6. Conclusions 110 6.1 Conclusions 110 6.2 Future Studies 113 Bibliography 115

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