关于次序统计量数学期望的两个恒等式的简证及其应用

给出了最大最小次序统计量分布函数和密度函数的恒等式,从而在随机变量独立同分布情形下,给出了关于次序统计量数学期望的两个恒等式的简捷证法.最后将几个恒等式应用于威布尔分布的最大次序统计量的分布及其数字特征的计算.     

一类最值问题的期望和方差

设X(n)=max(Xi),其中Xi(1≤i≤n)相互独立,均服从指数分布.利用指数分布的期望和二阶矩或通过讨论Xi的次序统计量,可求出X(n)的数学期望和方差.从两种不同解法可导出两个恒等式.     

离散型最小和最大次序统计量相关性研究

本文研究离散型随机变量之间的相关性度量, 讨论了最小次序统计量和最大次序统计量的渐近独立性, 给出了计算最小次序统计量和最大次序统计量的Kendall和Spearman秩相关系数的公式.     

多维Erlang混合的次序统计量及其在多生命理论中的应用（英文）

本文研究来自多维Erlang混合分布的具有相关性的随机变量的次序统计量并将相关结论应用到多生命理论中.利用熟知的数学归纳法推导出次序统计量分布密度函数的解析表达式并证明其分布为一维Erlang混合分布,寿险精算理论中常见的重要统计量可以算出精确解.这一结论将随机变量相互独立情形下的结果推广到变量存在相关性的情形.     

Fisher-Z分布次序统计量的普通多元随机序

本文研究了Fisher-Z分布次序统计量的随机比较问题.利用Beta随机变量的性质以及比例风险率模型的次序统计量随机比较的结论,获得了Fisher-Z分布次序统计量向量的普通多元随机序的比较,推广了文献中的相关结果.     

Archimedean copula刻画的尺度比例失效率模型的极小次序统计量的随机序北大核心CSCD

在多元链式优化序下,该文研究了两组来自于不同相依尺度比例失效率分布的最小次序统计量的随机比较.在某种数学意义下,一个由尺度比例失效率分布的不同脆弱参数和尺度参数构成的矩阵变化到另一个矩阵时,该文研究了在一定的条件下,来自于第一个尺度比例失效率分布的最小次序统计量在普通随机序下小于变化到的参数矩阵对应的尺度比例失效率分布的最小次序统计量.该文也给出了一些数值例子来说明得到的结果的正确性.     

关于n个独立同分布的指数分布的最值问题的期望和方差

对于X(n)=max1≤i≤n(Xi),其中Xi(1≤i≤n)独立同分布,均服从指数分布,求X(n)的数学期望和方差,本文给出了两种不同的解法,并且导出了两个恒等式.最后本文还从数学分析的角度证明了这两个恒等式.     

基于次序统计量的ES核估计

本文考虑期望损失ES(Expected Shortfall)的一个新的估计基于次序统计量的核估计,在α-混合序列条件下,给出它的Bahadur表达式和均方误差,并且证明该估计量的渐进正态性.     

关于离散型随机变量次序统计量分布矩阵对称性猜想的探索

本文研究了离散型随机变量次序统计量的分布矩阵的对称性 ,获得了二个定理 .定理 1　服从等概率二点分布或等概率三点分布的离散型随机变量的次序统计量的分布矩阵是对称矩阵 .定理 2　取值有限且等概率的离散型随机变量的次序统计量的分布矩阵具有中心对称性 .     

关于离散型随机变量次序统计量分布矩阵对称性猜想的探索

本研究了离散型随机变量次序统计量的分布矩阵的对称性,获得了二个定理。定理1服从等概率二点分布等概率三点分布的离散型随机变量的次序统计量的分布矩阵是对称矩阵。定理2取值有限且等概率的离散型随机变量的次序统计量的分布矩阵具有中心对称性。     

The Order of the Error Term for Moments of the Log Likelihood Ratio Unit Root Test in an Autoregressive Process

This paper investigates the asymptotics of the log likelihood ratio test for a unit root in an autoregressive (AR) process of general order. The main result is that the expectation and variance (in fact, all moments) of the test statistic may, to the order of T-1, where T is the number of observations, be approximated by the expectation and variance of the corresponding test in an AR(1) process. This result has obvious implications for the asymptotics of unit root tests for panels. An explicit formula for the approximation error of a test in an AR(2) process is also given.     

A rank test based on the moments of order statistics of the modified Makeham distribution

Single moments of order statistics from the modified Makeham distribution (MMD) are derived, an identity about the single moments of order statistics is given, and the specific expected value and variance of the single moments of order statistics from the MMD are calculated. In this study, the order statistic from the MMD was applied to the rank sum test in a two-sample problem. The exact critical values of the designated statistics were evaluated. Simulations were used to investigate the power of these statistics for the two-sided alternative with several population distributions. The powers of the statistics were compared with the Wilcoxon rank sum statistic, the Lepage statistic, the modified Baumgartner statistic, the Savage test and the normal score test. The Edgeworth expansion was used to evaluate the upper tail probability for the preferred statistic, given finite sample sizes.     

Stability of Order Statistics under Dependence

We precisely evaluate the upper and lower deviations of the expectation of every order statistic from an i.i.d. sample under arbitrary violations of the independence assumption, measured in scale units generated by various central absolute moments of the parent distribution of a single observation. We also determine the distributions for which the bounds are attained. The proof is based on combining the Moriguti monotone approximation of functions with the Hölder inequality applied for proper integral representations of expected order statistics in the independent and dependent cases. The method allows us to derive analogous bounds for arbitrary linear combinations of order statistics.     

离散型区间概率随机变量和模糊概率随机变量的数学期望

研究离散型区间概率随机变量和离散型第二类模糊概率随机变量数学期望的性质及求解方法．利用模糊分解定理,把求模糊概率随机变量的数学期望问题化为求一系列区间概率随机变量的数学期望．求区间概率随机变量的数学期望是一个典型的线性规划问题,用单纯形方法推导了求区间概率随机变量数学期望的一个很实用的计算公式．算例表明,用该计算公式得到的结果和用数学规划方法得到的结果完全吻合,但计算过程相对简单．     

On unbounded probability theory

The value of the empirical expectation coincides with that of the mean energy of an ideal Bose gas for one particle. The exact mathematical identity for these quantities makes it possible to carry over the concept of temperature corresponding to the mean energy to an unboundedly increasing sequence of random values for a new unbounded probability theory and for a generalization of Kolmogorov complexity theory. The notion of spectral gap, which was introduced in superconductivity theory, is carried over to unbounded probability theory.     

Properties of minimal mathematical expectations

The aim of this paper is to discuss how to compute a class of minimal mathematical expectations by using backward stochastic differential equations. We also prove that the minimal mathematical expectation operator still preserves some properties of the mathematical expectation operator.     

Phase-type distribution of customer relationship with Markovian response and marketing expenditure decision on the customer lifetime value

In recent years, more and more companies have adopted relationship marketing (RM). At the core of RM is the development and maintenance of long-term relationships with valuable customers. In RM, the customer lifetime value (CLV) is the discounted profit streams of a customer across the entire customer life cycle. The CLV plays a key role in customer acquisition and retention decisions. In this paper, we present a general mathematical framework for RM, and introduce a Markov chain model which is appropriate in modeling RM because of its flexibility and probabilistic nature. We also develop the life distribution of the customer relationship, which is a phase-type distribution since it is the distribution of the first arrival stopping state. And we obtain the expectation of the CLV, which is an important statistic for good decision-making. Finally, we illustrate how to find the optimal remarketing policy numerically. The framework developed for RM systems in this paper should be seen as a practical approach to RM where one can directly apply the results of phase-type distribution and expectation of CLV to marketing decisions.     

Expectations in normed 1-algebras

A positive hermitian linear functional on a ¹-algebra with identity is called an expectation when it satisfies a normalizing and an admissibility condition. The main theorem of this note shows that a positive linear function, on a normed ¹-algebra, with normalizing condition is always an expectation. A simple relation connecting this expectation and the expectation of Nakamura and Turumaru is given.     

On exponential scores statistics for testing against positive aging

It has been shown in this paper that a test proposed by Barlow and Doksum (1972) based on the exponential scores statistic for testing exponentiality against increasing failure rate distributions is consistent for the much wider class of harmonic new better than used in expectation distributions.