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Exponential and power-law probability distributions of wealth and income in the United King

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arXiv:cond-mat/0103544v2 [cond-mat.stat-mech] 28 Mar 2001Exponentialandpower-lawprobabilitydistributionsofwealthandincomeintheUnitedKingdomandtheUnitedStates

AdrianDr˘agulescu,VictorM.Yakovenko1DepartmentofPhysics,UniversityofMaryland,CollegePark,MD20742-4111,cond-mat/0103544,v.2March28,2001USA

1IntroductionThestudyofwealthandincomedistributionshasalonghistory.Pareto[1]proposedin1897thatwealthandincomedistributionsobeyuniversalpowerlaws.Subsequentstudieshaveoftendisputedthisconjecture(seeasystematicsurveyintheWorldBankresearchpublication[2]).Mandelbrot[3]proposedthattheParetolawappliesonlyasymptoticallytothehighendsofthedis-tributions.ManyresearcherstriedtodeducetheParetolawfromatheoryofstochasticprocesses.Gibrat[4]proposedin1931thatincomeandwealtharegovernedbymultiplicativerandomprocesses,whichresultinalog-normaldistribution.Theseideaswerelaterfollowed,amongmanyothers,byMon-trollandShlesinger[5].However,alreadyin1945Kalecki[6]pointedoutthatthelog-normaldistributionisnotstationary,becauseitswidthincreasesintime.Moderneconophysicists[7–9]alsousevariousversionsofmultiplicativerandomprocessesintheoreticalmodelingofwealthandincomedistributions.Unfortunately,numerousrecentpapersonthissubjectdoverylittleornocomparisonatallwithrealstatisticaldata,muchofwhichiswidelyavailablethesedaysontheInternet.Inordertofillthisgap,weanalyzedthedataonincomedistributionintheUnitedStates(US)fromtheBureauofCensusandtheInternalRevenueService(IRS)inRef.[10].Wefoundthattheindivid-ualincomeofabout95%ofpopulationisdescribedbytheexponentiallaw.Theexponentiallaw,alsoknowninphysicsastheBoltzmann-Gibbsdistribu-tion,ischaracteristicforaconservedvariable,suchasenergy.InRef.[11],wearguedthat,becausemoney(cash)isconserved,theprobabilitydistributionofmoneyshouldbeexponential.Wealthcanincreaseordecreasebyitself,butmoneycanonlybetransferredfromoneagenttoanother.So,wealthisnotconserved,whereasmoneyis.Thedifferenceisthesameasthedifferencebetweenunrealizedandrealizedcapitalgainsinstockmarket.Unfortunately,wewerenotabletofinddataonthedistributionofmoney.Ontheotherhand,wefounddataonwealthdistributionintheUnitedKing-dom(UK),whicharepresentedinthispaper.AlsopresentedaretheincomedistributiondatafortheUKandfortheindividualstatesoftheUSA.Inallofthesedata,wefindthatthegreatmajorityofpopulationisdescribedbyanexponentialdistribution,whereasthehigh-endtailfollowsapowerlaw.

2WealthdistributionintheUnitedKingdomInthissection,wediscussthecumulativeprobabilitydistributionofwealthN(w)=(thenumberofpeoplewhoseindividualwealthisgreaterthanw)/(thetotalnumberofpeople).AplotofNvs.wisequivalenttoaplotofper-son’srankinwealthvs.wealth,whichisoftenusedfortopreachpeople[12].Wewillusethepowerlaw,N(w)∝1/wα,andtheexponentiallawN(w)∝exp(−w/W),tofitthedata.Thesedistributionsarecharacterizedbytheexponentαandthe“temperature”W.Thecorrespondingprobabil-itydensities,P(w)=−dN(w)/dw,alsofollowapowerlaworanexponen-tiallaw.Fortheexponentiallaw,itisalsousefultodefinethetemperaturesW(2)(alsoknownasthemedian)andW(10)usingthebasesof1/2and1/10:N(w)∝(1/2)w/W(2)∝(1/10)w/W(10).Thedistributionofwealthisnoteasytomeasure,becausepeopledonotreporttheirtotalwealthroutinely.However,whenapersondies,allassetsmustbereportedforthepurposeofinheritancetax.Usingthesedataandanadjustmentprocedure,theBritishtaxagency,theInlandRevenue(IR),recon-structedwealthdistributionofthewholeUKpopulation.InFig.1,wepresentthe1996dataobtainedfromtheirWebsite[13].Theleftpanelshowsthecu-mulativeprobabilityasafunctionofthepersonaltotalnetcapital(wealth),whichiscomposedofassets(cash,stocks,property,householdgoods,etc.)andliabilities(mortgagesandotherdebts).Themainpanelillustratesinthe

21010010000.01%0.1%1%10%100% Total net capital (wealth), kpoundsCumulative percent of people

United Kingdom, IR data for 1996

ParetoBoltzmann−Gibbs

02040608010010%100%

Total net capital, kpounds0102030405060708090100%0

10

2030405060708090100%Cumulative percent of peopleCumulative percent of wealth

United Kingdom, IR data for 1996

Fig.1.Cumulativeprobabilitydistributionoftotalnetcapital(wealth)showninlog-log(leftpanel),log-linear(inset),andLorenz(rightpanel)coordinates.Points:theactualdata.Solidlines:leftpanel–fitstotheexponential(Boltzmann-Gibbs)andpower(Pareto)laws,rightpanel–function(2)calculatedfortheexponentiallaw.

log-logscalethatabove100k£thedatafollowapowerlawwiththeexpo-nentα=1.9.Theinsetshowsinthelog-linearscalethatbelow100k£thedataisverywellfittedbyanexponentialdistributionwiththetemperatureWUK=59.6k£(W(2)UK=41.3k£andW(10)UK=137.2k£).TherightpanelofFig.1showstheso-calledLorenzcurve[2,10]forwealthdistribution.Thehorizontalandverticalcoordinatesarethecumulativepop-ulationx(w)andthecumulativewealthy(w):

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随机微分方程在数理金融中的应用硕士学位论文 精品

随机微分方程在数理金融中的应用硕士学位论文 精品

摘要复杂数据主要表现在相依、非线性、维数高与不完全观测等，在股市、基因序列和经济等领域中经常出现。

为解决巨型数据集合问题，数据挖掘的理论、方法和技术已应运而生。

而针对诸如怎样同时检验成千上万个基因中哪些基因的表达水平有显著性差异之类的高维统计推断问题，以错误发现率为主要特征的非参数估计方法无疑为其提供了一个有效的解决途径。

本文主要研究考察错误发现率的在各种参数模型和非参数模型下的控制检验方法，全文共分为四章。

文章首先介绍了所选取课题的背景和意义，以及国内外在该方向的研究现状。

在多重假设检验的背景下，给出了错误发现率的定义，提出利用p值进行假设检验，并在假设检验独立和相依的情形下对错误发现率的控制方法进行了探讨。

在研究错误发现率的控制方法时，发现在处理多重假设检验问题时，核心的问题是如何估计真实零假设的个数，因此本文采用经验贝叶斯估计来估计它的值。

在参数混合模型和非参数混合模型中研究真实零假设的估计问题是本文的核心内容。

针对正态混合分布模型和Beta混合分布模型两种参数混合模型，文章采用矩估计方法和基于p值的最小二乘估计方法进行研究；在研究非参数混合模型时，分别介绍了最小二乘估计方法、Beta分布拟合模型和Beinstein 多项式拟合模型的方法。

文章的最后以Hedenfalk报告的一组乳腺癌患者的基因数据为例进行仿真研究，发现错误发现率为微阵列数据的多重假设检验提供了合适的错误控制指标。

关键词：错误发现率；多重假设检验；p值；非参数估计；微阵列数据- I -AbstractComplex data always appear in the stock market, gene sequences, economic and other fields, which mainly show the characteristic of dependent, nonlinear, high dimension and incomplete observations. In order to solve the problem of huge data collection, the theories, methods and techniques of data mining are proposed. While how to examine the high-dimensional statistical inference problem, such as the significant differences of expression levels in thousands of genes, the non-parametric estimation of false discovery rate provide an effective solution.This paper mainly investigate the test method based on the false discovery rate of various parametric model and non-parametric model, which is divided into four chapters. Firstly, this paper introduce the background and significance of the topic, and the current studies in this direction at home and abroad. Under the background of multiple hypotheses testing, the paper describe the definition of the false discovery rate, propose using the p-value to test the hypothesis testing, and discuss the controlling method of the false discovery rate when the hypotheses testing is independent or dependent. When we investigate the controlling method of the false discovery rate and studied the multiple hypothesis testing problem, we find that the central problem is how to estimate the number of true null hypothesis, so this paper use the empirical Bayes estimation to estimate its value. Investigating the estimation of true null hypothesis in the mixing parametric model and non-parametric model is core of the dissertation. Aiming at the mixed normal distribution model and Beta mixture distribution model, This paper use the method of moment estimation and least squares estimation method based on the p-value to estimate its value; On studying thenon-parametric mixture model, the paper introduce the least square estimation method, Beta distribution fitting model method and the Beinstein polynomial fitting model method. Finally, the paper conduct the simulation research based on a group of patients with breast cancer gene data by Hedenfalk, and find that the false discovery rate is able to provide a suitable error control targets for the multiple hypothesis testing of microarray data.Keywords: false discovery rate, multiple hypotheses testing, p-value, non-parametric estimation, microarray data- II -- III -目 录摘 要 ..................................................................................................................... I Abstract ................................................................................................................... I I第1章 绪 论 (1)1.1 课题研究的背景及意义 (1)1.2 国内外在该方向的研究现状 (1)1.2.1 国外对错误发现率的研究现状 (1)1.2.2 国内研究现状 (3)1.3 本文拟研究的主要内容 (3)1.4 创新点 (3)第2章 错误发现率的多重检验方法 (5)2.1 多重假设检验的错误测度 (5)2.2 P 值的定义、性质和计算方法 (6)2.3 独立情形下基于FDR 控制的检验方法 (7)2.4 相依情形下基于FDR 控制的检验方法 (8)2.5 真实零假设的个数0m 或比值0π的估计 (9)2.5.1 -λ估计 (9)2.5.2 经验贝叶斯估计 (11)2.6 本章小结 (12)第3章 参数混合模型和非参数混合模型的估计 (13)3.1 引言 (13)3.2 正态分布混合模型 (13)3.3 Beta 分布混合模型 (17)3.4 非参数混合模型的估计 (21)3.4.1 最小二乘估计 (22)3.4.2 Beta 分布拟合模型 (23)3.4.3 Beinstein 多项式拟合模型 (25)3.5 本章小结 (26)第4章 错误发现率的估计方法的应用 (27)4.1 引言 (27)4.2 微阵列数据实例研究 (27)4.3 本章小结 (28)结论 (30)参考文献 (31)哈尔滨工业大学学位论文原创性声明和使用权限 .......... 错误！未定义书签。

美国质量学会注册可靠性工程师CRE知识大纲

美国质量学会注册可靠性工程师CRE知识大纲

美国质量学会注册可靠性工程师(CRE)知识大纲The topics in this Body of Knowledge include additional detail in the form of subtext explanations and the cognitive level at which the questions will be written. This information will provide useful guidance for both the Examination Development Committee and the candidates preparing to take the exam. The subtext is not intended to limit the subject matter or be all-inclusive of what might be covered in an exam. It is intended to clarify the type of content to be included in the exam. The descriptor in parentheses at the end of each entry refers to the highest cognitive level at which the topic will be tested. A more comprehensive description of cognitive levels is provided at the end of this document.本知识大纲的内容中包含以文本阐释的形式提供的更多详情，还包括问题将被考核的认知水平。

本信息对考试发展委员会及准备考试的投考者都将提供有用的指导。

本文本决非旨在限制题目或包括考试中可能涉及的一切内容。

金融幂律

金融幂律

金融幂律：实证，模型以及原理Thomas Lux1Department of EconomicsUniversity of Kiellux@bwl.uni-kiel.de为社会科学中的幂律而准备：发现社会科学的复杂性和非均衡动态，Claudio Cioffi,编者.概述：金融市场（股票市场，外汇市场以及其他市场）均以一系列普遍使用的幂律为特征，最突出的例子是以强健的，近似三次幂律（cubic power law）来描述大量回报（large returns）的分布这一独特发现。

一个几乎同样强健特征是对波动性的长期依赖（例如：其自相关函数的双曲线递减）。

最近的文献加入了交易量的时间标度（temporal scaling）和higher moments of returns的multi-scaling. 最近，对于这些特征的关注引起了人们对市场过程中这些关键特征之所以出现的原因进行理论解释的尝试。

理论上来讲，不用种类的动态过程可能造成了这些幂律。

在经济学文献中找到的例子包括乘法随机过程（multiplicative stochastic processes）和多均衡的动态过程。

尽管这两种过程dynamics均以间歇产生大规模活动爆发的间断行为为特征，而事实上它们则可能是基于对交易过程上根本性地不同的理解。

前面的章节既回顾了在经济学文献中提出的相关模型，又复习了出现在以上数据到产生机制中的幂律的分析背景。

一简介尽管关于收入和资产中的幂律研究可以追溯到19世纪（Pareto）, 但对于在金融数据中幂律的关注还是最近的事情。

幂律在金融中第一次体现可能是在Mandelbrot 的Journal of Business 中1963卷的具有开启意义的Variation of Certain Speculative Prices"一文中，随后紧接着Eugene Fama在随后的一篇名为“Mandelbrot and the Stable Paretian Hy-pothesis"的一文中，进一步阐述了同一个问题。

meijerg函数 计算例子

meijerg函数 计算例子

meijerg函数计算例子Meijer G-function: Calculation ExamplesThe Meijer G-function, introduced by Cornelius Lanczos, is a special function in mathematics that generalizes many other types of special functions, such as the hypergeometric function. It has applications in various branches of mathematics and physics, including number theory, probability theory, and quantum mechanics.To better understand the Meijer G-function, let's consider a few calculation examples:Example 1: Computing the Meijer G-function for a simple caseSuppose we want to calculate the value of the Meijer G-function G_{1, 2}^{2,1}\left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \:|\: z\right), where z is a complex variable. This function can be expressed as an integral of a product of exponential and polynomial functions. By properly evaluating the integral and simplifying the expression, we can find the numerical value of the Meijer G-function for any given z.Example 2: Evaluation of Meijer G-function for specific parametersConsider the Meijer G-function G_{2, 3}^{3, 2}\left(\begin{smallmatrix} 1, 2 \\ 2, 3, 4 \end{smallmatrix} \:|\: z\right). In this example, we have specific parameters for the G-function. We can substitute the values into the G-function and solve it using numerical techniques such as series expansion or optimization algorithms. The result will provide us with the numerical value of the G-function for the given parameters and z.Example 3: Applications of Meijer G-function in probability theoryOne significant use of the Meijer G-function is in probability theory. For instance, when dealing with probability density functions (PDFs) or cumulative distribution functions (CDFs) for certain statistical distributions, the Meijer G-function can be employed to express these functions efficiently and accurately. By utilizing the propertiesand formulas associated with the Meijer G-function, researchers can analyze the behavior and properties of various probability distributions in a more general framework.Example 4: Meijer G-function in quantum mechanicsIn the field of quantum mechanics, the Meijer G-function finds applications in solving differential equations and characterizing the behavior of wavefunctions. When dealing with systems that exhibit special symmetry or non-trivial potential profiles, the Meijer G-function can be utilized to express the solutions for specific energy levels or eigenfunctions. This enables physicists to better understand the quantum behavior of particles in complex systems.These examples demonstrate how the Meijer G-function can be used to solve mathematical problems and analyze physical phenomena in different areas of study. It is a powerful tool that allows mathematicians, physicists, and engineers to tackle complex calculations and gain deeper insights into the behavior of various systems.Remember, the examples mentioned here are just a glimpse into the wide-ranging applications of the Meijer G-function. Its versatility and usefulness make it an essential tool for mathematical modeling and analysis in diverse fields.。