針對負二項分配參數建立一些信賴區間之研究

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。

關鍵字

負二項分配；包含機率；期望長度；信賴區間；貝氏可靠區間

並列摘要

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.

並列關鍵字

Negative binomial parameter ； Coverage probability ； Expected length ； Confidence Intreval ； Bayes credible

參考文獻

[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

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針對負二項分配參數建立一些信賴區間之研究

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