本文選題：排序集抽樣 + 伽馬分布��； 參考：《吉首大學》2017年碩士論文

【摘要】：伽馬分布是概率論與數(shù)理統(tǒng)計中非常重要的一種分布,其應用非常廣泛,尤其在水文學、可靠性理論、壽險精算等領域.因此,廣泛受到國內(nèi)外學者專家的關注,而研究伽馬分布的參數(shù)估計是其一個重要內(nèi)容.廣大學者專家借助傳統(tǒng)的簡單隨機抽樣(SRS)選取樣本,使用矩估計、區(qū)間估計、極大似然估計等方法來研究伽馬分布的參數(shù)估計.由于簡單隨機抽樣局限性使得選取的樣本代表性不是很強,所以排序集抽樣(RSS)應運而出.RSS于1952年被Mc Intyre最先提出并用于估計某農(nóng)場的產(chǎn)量.這種抽樣方法在估計同一個總體時所需要的樣本容量比簡單隨機抽樣更少,在相同樣的本容量下,由RSS得到的樣本包含了更多的總體信息.使得RSS要比傳統(tǒng)的SRS獲取數(shù)據(jù)更為有效.因此,RSS廣泛受到國內(nèi)外廣大學者專家的青睞,得到蓬勃發(fā)展.本文便是使用排序集抽樣抽取樣本來研究伽馬分布參數(shù)的極大似然估計(MLE).本文在第一章介紹了用RSS研究MLE的背景、相關理論及發(fā)展狀態(tài).然后給出要研究的主要內(nèi)容:一是研究了伽馬分布在RSS下刻度參數(shù)的MLE,并對其的存在性及唯一性給出了理論證明;二是研究了伽馬分布在RSS下形狀參數(shù)的MLE,同時也給出了形狀參數(shù)MLE的存在性及唯一性的理論證明,接下來提出了一種新的抽樣方式——基于Fisher信息量最大化排序集抽樣并用于研究伽馬分布形狀參數(shù)的MLE;三是研究了伽馬分布在刻度參數(shù)和形狀參數(shù)均未知時的刻度參數(shù)及形狀參數(shù)的MLEs,并對MLE的存在性給出了理論證明;四是對每一種情形下的極大似然估計進行數(shù)值模擬并與簡單隨機抽樣下的MLE進行對比,得出RSS下的參數(shù)估計比簡單隨機抽樣下參數(shù)的MLE效果更好,均方誤差更小.最后總結全文,并對未來的研究進行展望.
[Abstract]:Gamma distribution is a very important distribution in probability theory and mathematical statistics. Its application is very extensive, especially in the fields of hydrology, reliability theory, life insurance actuarial and so on. Therefore, many scholars and experts at home and abroad pay close attention to it, and the parameter estimation of gamma distribution is an important part of it. The majority of scholars and experts use the traditional simple random sampling (SRS) to select samples and use the methods of moment estimation, interval estimation and maximum likelihood estimation to study the parameter estimation of gamma distribution. Because of the limitation of simple random sampling, the selected samples are not very representative, so the ordered set sampling (RSS) should be shipped out. RSS was first proposed by MC Intyre in 1952 and used to estimate the yield of a farm. This sampling method requires less sample size than simple random sampling when estimating the same population. Under the same capacity, the sample obtained by RSS contains more information on the whole population. RSS is more efficient than traditional SRS in getting data. Therefore, RSS has been widely favored by domestic and foreign scholars and experts, and developed vigorously. In this paper, the maximum likelihood estimation of gamma distribution parameters is studied by sampling samples from ordered sets. In the first chapter, we introduce the background, theory and development of MLE with RSS. The main contents of this paper are as follows: first, we study the scale parameter of gamma distribution under RSS, and prove its existence and uniqueness in theory. The second is to study the shape parameter of gamma distribution under RSS, and to prove the existence and uniqueness of shape parameter MLE. Then a new sampling method is proposed, which is based on Fisher information maximization sorting set sampling and is used to study the shape parameters of gamma distribution. Third, we study the engraving of gamma distribution when the calibration parameters and shape parameters are unknown. MLEs of degree parameter and shape parameter, and the existence of MLE is proved theoretically. Fourth, the maximum likelihood estimation in each case is numerically simulated and compared with the MLE under simple random sampling. It is concluded that the parameter estimation under RSS is more effective than the MLE under simple random sampling, and the mean square error is smaller. Finally, the full text is summarized, and the future research is prospected.
【學位授予單位】：吉首大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O212.1

【參考文獻】

相關期刊論文 前4條

1 魯春林;方東輝;陳望學;錢文舒;;基于遺傳算法Beta分布參數(shù)的極大似然估計[J];吉首大學學報(自然科學版);2016年05期

2 陳望學;謝民育;劉佳瑩;周q;;排序集下單指數(shù)分布均值的修正極大似然估計[J];華中師范大學學報(自然科學版);2013年06期

3 黃華;宋艷萍;趙磊;;伽瑪分布參數(shù)的極大似然估計數(shù)值解法[J];高等函授學報(自然科學版);2011年05期

4 王玉,張星,韋卓信;確定防洪堤設計水位的方法探討[J];廣西水利水電;1999年03期

相關碩士學位論文 前1條

1 陳望學;動態(tài)排序集抽樣下刻度分布族刻度參數(shù)的參數(shù)估計[D];華中師范大學;2012年

，

本文編號：1851696