基于排序集抽样下伽马分布参数的极大似然估计

发布时间：2018-05-06 09:08

本文选题：排序集抽样 + 伽马分布　；参考：《吉首大学》2017年硕士论文

【摘要】：伽马分布是概率论与数理统计中非常重要的一种分布,其应用非常广泛,尤其在水文学、可靠性理论、寿险精算等领域.因此,广泛受到国内外学者专家的关注,而研究伽马分布的参数估计是其一个重要内容.广大学者专家借助传统的简单随机抽样(SRS)选取样本,使用矩估计、区间估计、极大似然估计等方法来研究伽马分布的参数估计.由于简单随机抽样局限性使得选取的样本代表性不是很强,所以排序集抽样(RSS)应运而出.RSS于1952年被Mc Intyre最先提出并用于估计某农场的产量.这种抽样方法在估计同一个总体时所需要的样本容量比简单随机抽样更少,在相同样的本容量下,由RSS得到的样本包含了更多的总体信息.使得RSS要比传统的SRS获取数据更为有效.因此,RSS广泛受到国内外广大学者专家的青睐,得到蓬勃发展.本文便是使用排序集抽样抽取样本来研究伽马分布参数的极大似然估计(MLE).本文在第一章介绍了用RSS研究MLE的背景、相关理论及发展状态.然后给出要研究的主要内容:一是研究了伽马分布在RSS下刻度参数的MLE,并对其的存在性及唯一性给出了理论证明;二是研究了伽马分布在RSS下形状参数的MLE,同时也给出了形状参数MLE的存在性及唯一性的理论证明,接下来提出了一种新的抽样方式——基于Fisher信息量最大化排序集抽样并用于研究伽马分布形状参数的MLE;三是研究了伽马分布在刻度参数和形状参数均未知时的刻度参数及形状参数的MLEs,并对MLE的存在性给出了理论证明;四是对每一种情形下的极大似然估计进行数值模拟并与简单随机抽样下的MLE进行对比,得出RSS下的参数估计比简单随机抽样下参数的MLE效果更好,均方误差更小.最后总结全文,并对未来的研究进行展望.
[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.
【学位授予单位】：吉首大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1

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