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System No. U0002-2706200817331700
Title (in Chinese) 針對負二項分配參數建立一些信賴區間之研究
Title (in English) A study of Some of the Confidence Intervals for a Negative Binomial Parameter
Other Title
Institution 淡江大學
Department (in Chinese) 管理科學研究所碩士班
Department (in English) Graduate Institute of Management Science
Other Division
Other Division Name
Other Department/Institution
Academic Year 96
Semester 2
PublicationYear 97
Author's name (in Chinese) 劉宗翰
Author's name(in English) Tsung-Han Liu
Student ID 695620566
Degree 碩士
Language Traditional Chinese
Other Language
Date of Oral Defense 2008-06-09
Pagination 41page
Committee Member advisor - Kuo-Ren Lou
co-chair - 楊維楨
co-chair - 徐政義
Keyword (inChinese) 負二項分配
包含機率
期望長度
信賴區間
貝氏可靠區間
Keyword (in English) Negative binomial parameter
Coverage probability
Expected length
Confidence Intreval
Bayes credible
Other Keywords
Subject
Abstract (in Chinese)
此篇文章是針對負二項分配參數p做區間估計,是「包含機率」(Coverage Probability),及「信賴區間長度」(Confidence Interval Expected Length),作為評量的一個標準,利用這兩個方法來比較及挑選出何種信賴區間的估計方法最符合我們的要求,期待使包含機率高且期望長度短。  
    一直以來,研究學者多是研究二項分配參數的信賴區間為多,在負二項分配參數信賴區間的著墨則較少,則本論文則使用Clopper-Pearson的方法,利用這方法來找出參數p的信賴區間,比較在不同參數下,評估這些參數呈現結果的優缺點。
    本文也加入貝氏信賴區間的探討,各做了以無訊息先驗分配(non-informative Prior distribution)及有訊息先驗分配(informative Prior distribution)以貝它(Beta)為先驗的貝氏區間估計。其中以貝它為先驗的分配,做了參數α,β為(a)α=1/2,β=1/2(b)α=1,β=1/2(c)α=5,β=1/2(d)α=10,β=1/2四種的數值模擬。比較出何者呈現的結果最能符合標準。
Abstract (in English)
This article is providing confidence intervals for a negative binomial 
distribution using Coverage Probability and Confidence Interval Expected Length to be criteria which pick out whose Confidence Interval Expected Length is shorter and Coverage Probability is higher that reach our requirements.
    Many researchers investigated confidence interval for binomial distribution but few researchers studied confidence interval for a negative binomial distribution. We use the method of Clopper-Pearson to find whether parameter p of a negative binomial distribution is in its interval and compare with two different parameters which one is better.
    This article also adds Bayes methods that include non-informative Prior distribution and informative Prior distribution. Therefore, we consider 4 cases of Beta distribution for Prior as follow (a)α=1/2,β=1/2(b)α=1,β=1/2(c)α=5,β=1/2(d)α=10,β=1/2. We try to find a appropriate confidence interval to reach confidence level (1-α) or higher with small sample size.
Other Abstract
Table of Content (with Page Number)
目錄
目錄Ⅰ
表目錄Ⅲ
圖目錄Ⅳ
第一章  緒論1
1.1 研究動機與目的1
1.2 文獻回顧2
1.3 研究架構4
第二章  負二項分配參數的信賴區及模擬結果5
2.1 負二項分配參數的(Clopper-Pearson)信賴區間估計5
2.2 包含機率與信賴區間期望長度6
2.3 數值模擬結果8
第三章  負二項分配參數貝氏可靠區間14
3.1  負二項分配參數的先驗分配為費雪(Fisher)貝氏區間估計14
3.2  費雪(Fisher)貝氏可靠區間數值模擬結果16
3.3  負二項分配參數的先驗分配為貝它(Beta)貝氏區間估計18
3.4  貝它(Beta)貝氏可靠區間數值模擬結果20
第四章  結論與建議32
4.1  研究結論32
4.2  後續研究之建議34
參考文獻35
附錄37
附錄一37
附錄二40
表目錄
表3-1:負二項分配樣本數n=10,r=1,在信賴水準95%下,貝氏(先
      驗分配各為費雪、貝它分配)可靠區間於不同參數p情形下
      的包含機率值表28
表3-2:負二項分配樣本數n=10,r=1,在信賴水準95%下,Clopper-
      Pearson信賴區間和貝氏(先驗分配各為費雪、貝它分配)可靠
      區間於不同參數p情形下的包含機率值表30
圖目錄
圖2-1:負二項分配在參數p=0.1,r=1,信賴水準為95%之下Clopper-Pearson的包含機率圖8
圖2-2:負二項分配在參數p=0.5,r=1,信賴水準為95%之下Clopper-Pearson的包含機率圖8
圖2-3:負二項分配在參數p=0.1,r=1,信賴水準為95%之下Clopper-Pearson的期望長度圖10
圖2-4:負二項分配在參數p=0.5,r=1,信賴水準為95%之下Clopper-Pearson的期望長度圖10
圖2-5:負二項分配在樣本數n=10、r=1、信賴水準95%之下,畫出  
      Clopper-Pearson信賴區間估計在不同參數p值的包含機率圖12
圖2-6:負二項分配在樣本數n=10、r=1、信賴水準95%之下,畫出
      Clopper-Pearson信賴區間估計在不同參數p值的期望長度圖12
圖3-1:負二項分配在參數p=0.1,r=1,信賴水準95%之下,貝氏可靠 
      區間於不同樣本數下的機率圖16
圖3-2:負二項分配在參數p=0.5,r=1,信賴水準95%之下,貝氏可靠 
      區間於不同樣本數下的機率圖16
圖3-3:負二項分配在樣本個數n=10,r=1,信賴水準95%之下,貝氏可
      靠區間於不同參數p下的機率圖17
圖3-4:負二項分配在參數α=1/2,β=1/2,p=0.1,r=1,信賴水準
      95%之下,貝氏可靠區間於不同樣本數下的機率圖20
圖3-5:負二項分配在參數α=1/2,β=1/2,p=0.5,r=1,信賴水準
      95%之下,貝氏可靠區間於不同樣本數下的機率圖20
圖3-6:負二項分配在參數α=1,β=1/2,p=0.1,r=1,信賴水準95%
      之下,貝氏可靠區間於不同樣本數下的機率圖21
圖3-7:負二項分配在參數α=1,β=1/2,p=0.5,r=1,信賴水準95%
      之下,貝氏可靠區間於不同樣本數下的機率圖21
圖3-8:負二項分配在參數α=5,β=1/2,p=0.1,r=1,信賴水準95%
      之下,貝氏可靠區間於不同樣本數下的機率圖22
圖3-9:負二項分配在參數α=5,β=1/2,p=0.5,r=1,信賴水準95%
      之下,貝氏可靠區間於不同樣本數下的機率圖22
圖3-10:負二項分配在參數α=10,β=1/2,p=0.1,r=1,信賴水準 
       95%之下,貝氏可靠區間於不同樣本數下的機率圖23
圖3-11:負二項分配在參數α=10,β=1/2,p=0.5,r=1,信賴水準
       95%之下,貝氏可靠區間於不同樣本數下的機率圖23
圖3-12:負二項分配在參數α=1/2,β=1/2,樣本個數n=10,r=1,
       信賴水準95%之下,貝氏可靠區間於不同參數p下的機率圖24
圖3-13:負二項分配在參數α=1,β=1/2,樣本個數n=10,r=1,信賴
       水準95%之下,貝氏可靠區間於不同參數p下的機率圖24
圖3-14:負二項分配在參數α=5,β=1/2,樣本個數n=10,r=1,信賴
       水準95%之下,貝氏可靠區間於不同參數p下的機率圖25
圖3-15:負二項分配在參數α=10,β=1/2,樣本個數n=10,r=1,信
       賴水準95%之下,貝氏可靠區間於不同參數p下的機率圖25
圖3-16:負二項分配在樣本個數n=10,r=1,信賴水準95%之下,貝氏 
       可靠區間於不同參數p情況的期望長度圖28
圖3-17:負二項分配樣本數n=10,r=1,在信賴水準95%下,Clopper-
       Pearson信賴區間和貝氏(先驗分配各為費雪、貝它分配)可靠
        區間於不同參數p情形下的期望長度圖30
References
[1] Berger,James O. (1985), Statistical Decision Theory and Bayesian Analysis Second Edition, Spring-Verlag New York Heidelberg Tokyo

[2] Blaker,Helge (2000), Confidence Curves and Improved Exact Confidence Intervals for Discrete Distributions, The Canadian Journal of Statistics / La Revue Canadienne de Statistique, Vol. 28, No. 4, pp. 783-798 

[3] Bohning,Dankmar (1994), Better Approximate Confidence Intervals for a Binomial Parameter, The Canadian Journal of Statistics / La Revue Canadienne de Statistique, Vol. 22, No. 2, pp. 207-218 

[4] Byrne,John (2005), A New Short Exact Geometric Confidence Interval,Australian & New Zealand Journal of Statistics, 47 (4), PP.563–569, 

[5] Henderson,Michael and Meyer,Mary C. (2001), Exploring the Confidence Interval for a Binomial Parameter in a First Course in Statistical Computing, The American Statistician ,Vol. 55, No. 4 ,pp. 337-344

[6] Howden,W. (1997), Confidence-based reliability and statistical coverage estimation. Proc. Int. Symp. Software Reliability Engineering, pp. 283–291

[7] Krishnamoorthy, K. (2006), Handbook of Statistical Distribution with Applications, Chapman&Hall/CRC

[8] Lu,Wang-shu (2000), Improve Confidence Intervals for a Binomail Parameter Using The Bayesian Method, Cummun.Statist.-Theory Method , 29(12),PP. 2835-2847

[9]  Lui,Kung-Jong (1995), Confidence limits for the population prevalence rate based on the negative binomial distribution, Statistics in Medicine, Vol.14 , pp.1471-1477

[10] Sahinoglu,M. (2003), An Empirical Bayesian Stopping Rule in Testing and Verification of Behavioral Models, IEEE Trans, Instrumentation and Measurement, Vol. 52, No. 5, pp. 1428–1443

[11] Samuels,Myra L. and Lu,Tai-Fang C. (1992), Sample Size 
Requirements for the Back-of-the-Envelope Binomial 
Confidence Interval, The American Statistician, Vol.46, No. 3, pp. 228-231
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