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

【摘要】：伽馬分布是概率論與數(shù)理統(tǒng)計(jì)中非常重要的一種分布,其應(yīng)用非常廣泛,尤其在水文學(xué)、可靠性理論、壽險(xiǎn)精算等領(lǐng)域.因此,廣泛受到國(guó)內(nèi)外學(xué)者專(zhuān)家的關(guān)注,而研究伽馬分布的參數(shù)估計(jì)是其一個(gè)重要內(nèi)容.廣大學(xué)者專(zhuān)家借助傳統(tǒng)的簡(jiǎn)單隨機(jī)抽樣(SRS)選取樣本,使用矩估計(jì)、區(qū)間估計(jì)、極大似然估計(jì)等方法來(lái)研究伽馬分布的參數(shù)估計(jì).由于簡(jiǎn)單隨機(jī)抽樣局限性使得選取的樣本代表性不是很強(qiáng),所以排序集抽樣(RSS)應(yīng)運(yùn)而出.RSS于1952年被Mc Intyre最先提出并用于估計(jì)某農(nóng)場(chǎng)的產(chǎn)量.這種抽樣方法在估計(jì)同一個(gè)總體時(shí)所需要的樣本容量比簡(jiǎn)單隨機(jī)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(guó)內(nèi)外廣大學(xué)者專(zhuān)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(lái)研究伽馬分布參數(shù)的極大似然估計(jì)(MLE).本文在第一章介紹了用RSS研究MLE的背景、相關(guān)理論及發(fā)展?fàn)顟B(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(duì)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(shí)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(lái)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(shí)的刻度參數(shù)及形狀參數(shù)的MLEs,并對(duì)MLE的存在性給出了理論證明;四是對(duì)每一種情形下的極大似然估計(jì)進(jìn)行數(shù)值模擬并與簡(jiǎn)單隨機(jī)抽樣下的MLE進(jìn)行對(duì)比,得出RSS下的參數(shù)估計(jì)比簡(jiǎn)單隨機(jī)抽樣下參數(shù)的MLE效果更好,均方誤差更小.最后總結(jié)全文,并對(duì)未來(lái)的研究進(jìn)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(xué)位授予單位】：吉首大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O212.1

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本文編號(hào)：1851696