Entropy和Extropy的统计性质研究

发布时间：2018-02-27 11:21

本文关键词： 熵 Extropy 序统计量记录值剩余分位熵剩余Extropy 随机序均匀性检验　出处：《中国科学技术大学》2017年博士论文　论文类型：学位论文

【摘要】：许多学者在关注如何度量一个随机变量包含的不确定性.文献Shannon (1948)中定义的香农熵是其中一个重要的度量.最近,Lad et al. (2015)提出一个新的度量,称作Extropy. 一个随机变量的Extropy可视为该随机变量香农熵的补对偶.随机变量的香农熵和Extropy不仅具有迷人的性质,而且也有广泛的应用,本论文对这两个概念做进一步研究.首先研究剩余分位熵.我们证明:递减平均剩余寿命类包含于递减剩余分位熵类,递减剩余分位熵类在混合运算下不封闭.剩余分位熵序在加速寿命模型和广义序统计量模型中封闭.另外,我们也研究了诸如IFR,IFRA,NBU类寿命分布的熵或者剩余分位熵的上界.其次探讨序统计量和记录值的Extropy性质.为此,先证明:随机变量之间色散序关系可推出各自序统计量、记录值之间的Extropy序关系.作为此结论的推论,建立了一些寿命分布类Extropy的上界.接着,我们展示,序统计量和记录值的Extropy可以唯一确定总体的分布.我们对序统计量和记录值的Extropy的单调性也进行了探讨.对具有DFR(IRHR)性质的总体,我们发现,序统计量的Extropy关于样本容量递减(递增),也关于序统计量的阶数递增(递减).对密度分位函数递增的总体,我们发现,记录值的Extropy关于样本容量递减.最后,我们还建立了序统计量和记录值的Extropy的下界.对密度对称的总体,证明了其序统计量的Extropy关于样本中位数对称.接着讨论密度未知的连续随机变量Extropy的估计问题.为此,我们构造了两个弱相合估计,并通过实例和蒙特卡洛模拟,研究了这两个估计量的(根)均方误差.然后,基于第二个估计量,我们对均匀性检验问题提出了一种新的检验方法.对这种新的检验方法,我们不仅给出了部分检验水平的临界值,还把我们检验方法的势值与其他检验方法的势值进行了比较.比较结果发现,我们的检验方法在很多情况下都不比其他方法差.最后研究连续随机变量的剩余Extropy问题.首先发现:连续随机变量的剩余Extropy可由其失效率函数唯一确定.基于此结论,对有限范围分布,指数分布,Pareto分布分别进行了等价刻画.其次发现:总体的分布可由序统计量的剩余Extropy唯一确定.另外,还研究了一阶序统计量剩余Extropy的单调性问题.基于剩余Extropy,定义了两个新的分布类,研究了这两个分布类的一些基本性质.为比较两个随机变量剩余Extropy的大小,定义了一个称之为剩余Extropy序的新随机序,并研究了剩余Extropy序与已有随机序之间的联系,以及在一些模型中的应用.
[Abstract]:Many scholars are concerned about how to measure the uncertainty contained in a random variable. Shannon entropy, as defined in reference Shannon / 1948, is one of the most important measures. Called Extropy.The Extropy of a random variable can be regarded as a complementary dual of the Shannon entropy of the random variable. The Shannon entropy and Extropy of the random variable not only have fascinating properties, but also have extensive applications. In this paper, we further study these two concepts. First, we study the residual quantile entropy. We prove that the reduced mean residual life class is contained in the decreasing residual quantile entropy class. The reduced residual quantile entropy class is not closed in the mixed operation. The residual quantile entropy order is closed in the accelerated life model and the generalized order statistics model. We also study the upper bound of entropy or residual entropy of NBU class life distribution such as IFRX IFRAN NBU. Secondly, we discuss the Extropy properties of order statistics and recorded values. For this reason, we first prove that the dispersion order relation among random variables can deduce their order statistics. As a corollary of this conclusion, the upper bound of some life distribution class Extropy is established. Then, we show that, The Extropy of order statistics and recorded values can uniquely determine the distribution of the population. We also discuss the monotonicity of Extropy of order statistics and recorded values. The Extropy of an ordered statistic is about the decrease of sample size (increments) and the order of ordered statistics (decrement). For the population with increasing density quantile function, we find that the Extropy of the recorded value is decreasing with respect to the sample size. Finally, We also establish the lower bound of Extropy for ordered statistics and recorded values. For the population with symmetric density, we prove that the Extropy of ordered statistics is symmetric with respect to the sample median. Then we discuss the estimation problem of continuous random variables Extropy with unknown density. We construct two weakly consistent estimators and study the (root) mean square error of the two estimators by examples and Monte Carlo simulations. Then, based on the second estimator, We propose a new test method for homogeneity test problem. For this new test method, we not only give the critical value of partial test level, The potential value of our test method is also compared with that of other test methods. Our test method is not inferior to other methods in many cases. Finally, we study the residual Extropy problem of continuous random variables. First, we find that the residual Extropy of continuous random variables can be uniquely determined by its failure rate function. For the finite range distribution, the exponential distribution and the Pareto distribution are characterized respectively. Secondly, it is found that the distribution of the population can be determined only by the residual Extropy of the ordered statistics. In this paper, we also study the monotonicity of residual Extropy, a first-order statistic. Based on the residual extropy, two new distribution classes are defined, and some basic properties of these two classes are studied. In order to compare the size of residual Extropy between two random variables, A new random order called residual Extropy order is defined, and the relation between residual Extropy order and existing random order is studied, as well as its application in some models.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O212

【相似文献】