Pareto Density Distributions

Building on new insights into the genesis ofPareto-Distributions,(“Kopp” effect etc.) as publishedearlier in “Quality and Quantity”, the author gives at least oneauthentic/definitive Pareto-Formula. A practical example of the synthetic generation of Pareto Distributions by means of spreadsheets. A working D.I.Y-method for fine-fitting Pareto-curvesto scattergrams with spreadsheets using interalia an indirect method of the least squares of residuals is fully demonstrated. A comparative test-fit to a cumulative Pareto- Distribution example, where a simulative curve-formula evolved by Prof. B. Arnold/Ucla is used for demonstration. Easy to absorb and to retain graphical tableaux are employed to visualize the chain of descent and interconnections between normal distributions, log-normal distributions and Pareto- Distributions. A quasi-dichotomy of the Pareto-formulae is presented in tableau-form. One innovative formula for Pareto-distribution is given as: F(x)= k^*e― [((ln(Integral(In(x)))) ‐ (ln(Integral(ln(μ)))))² / 2^*(ln(Integral(ln(σ))))²} Readers e-mailed constructive opinions &/or inputs are encouraged and welcomed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Connecting Distributions with Power Tails on the Real Line, the Half Line and the Interval

Univariate continuous distributions have three possible types of support exemplified by: the whole real line , , the semi‐finite interval and the bounded interval (0,1). This paper is about connecting distributions on these supports via ‘natural’ simple transformations in such a way that tail properties are preserved. In particular, this work is focussed on the case where the tails (at ±∞) of densities are heavy, decreasing as a (negative) power of their argument; connections are then especially elegant. At boundaries (0 and 1), densities behave conformably with a directly related dependence on power of argument. The transformation from (0,1) to is the standard odds transformation. The transformation from to is a novel identity‐minus‐reciprocal transformation. The main points of contact with existing distributions are with the transformations involved in the Birnbaum–Saunders distribution and, especially, the Johnson family of distributions. Relationships between various other existing and newly proposed distributions are explored.     

浅析二项分布、泊松分布和正态分布之间的关系

二项分布、泊松分布和正态分布一直是学习和研究概率统计的基础。在一定条件下,这三个分布之间存在着密切关系。文章通过求极限分布,研究了二项分布与泊松分布、二项分布与正态分布之间的关系,并利用特征函数和分布函数相互唯一确定这一性质,分析了泊松分布和正态分布之间的关系。     

Distributions slanted to the right

We solve Problem 234 in Statistica Neerlandica by introducing the concept of slantedness. Distributions with a decreasing Lebesgue density are slanted to the right. This is no longer true for distributions on a lattice with decreasing density. Both kinds of distributions have positive central odd moments.     

A bivariate Poisson count data model using conditional probabilities

The applied econometrics of bivariate count data predominantly focus on a bivariate Poisson density with a correlation structure that is very restrictive. The main limitation is that this bivariate distribution excludes zero and negative correlation. This paper introduces a new model which allows for a more flexible correlation structure. To this end the joint density is decomposed by means of the multiplication rule in marginal and conditional densities. Simulation experiments and an application of the model to recreational data are presented.     

On a study of the exponential and Poisson characteristics of the Poisson process

Based on the exponential and Poisson characteristics of the Poisson process, in this work we present some characterizations of the Poisson process as a renewal process. More precisely, let γ_t be the residual life at time t of the renewal process A={A(t),t≥0 }, under suitable condition, we prove that if Var(γ_t)=E ² (γ_t),∀t≥0, then A is a Poisson process. Secondly, we show that if Var (A(t)) is proportional to E (A(t)), then A is a Poisson process also, and Var (A(t))=E (A(t)). Received: August 1999     

泊松积分在概率论中的应用

文章介绍了泊松积分在正态分布、二维正态分布、对数正态分布等问题中常见的应用,并给出了具体的例子。     

泊松积分在概率论中的应用

文章介绍了泊松积分在正态分布、二维正态分布、对数正态分布等问题中常见的应用,并给出了具体的例子。     

基于泊松分布的兰州冻品市场面积估算

预测在我们的日常生活中起着非常作用,长久以来人们总结并归纳出的定性定量的预测方法很多。文章根据调查方法的特殊性用运概率中的离散和连续型概率方法分别对调查数据进行拟合,进而找到符合数据分布的分布规律并进行预测。     

基于收入分布的收入差距扩大成因的分解

本文采用核平滑化的半参方法估计了1987年、1996年、2004年我国城市居民收入分布曲线,并且基于收入分布又进一步分解了导致城市居民收入差距扩大的主要原因。收入差距扩大的成因可以分为两类:一是劳动者的劳动力特点的普遍变化,如劳动者教育水平的提高,行业、职业分布的变化;二是对劳动力特点回报率的变化,如教育回报率的提高,各行业、职业回报率差异的扩大等。本文发现后者是导致收入差距扩大的主要原因。     

Dispersion of growth paths of macroeconomic models in thermodynamic limits: two-parameter Poisson–Dirichlet models

This paper discusses dispersion of growth patterns of macroeconomic models in thermodynamic limits. More specifically, the paper shows that the coefficients of variations of the total numbers of clusters and the numbers of clusters of specific sizes of one- and two-parameter Poisson–Dirichlet models behave qualitatively differently in the thermodynamic limits. The coefficients of variations of the numbers of clusters in the former class of distributions are all self-averaging, while the those in the latter class are all non-self averaging. In other words, dispersions or variations of growth rates about the means do not vanish in the two-parameter version of the model, while they do in the one-parameter version in the thermodynamic limits. The paper ends by pointing out other models, such as triangular urn models, may converge to Mittag–Leffler distributions which exhibit non-self-averaging behavior for certain parameter combinations. The author is grateful for many helps he received from H. Yoshikawa, and M. Sibuya.     

Robust Likelihood Methods Based on the Skew-t and Related Distributions

The robustness problem is tackled by adopting a parametric class of distributions flexible enough to match the behaviour of the observed data. In a variety of practical cases, one reasonable option is to consider distributions which include parameters to regulate their skewness and kurtosis. As a specific representative of this approach, the skew‐t distribution is explored in more detail and reasons are given to adopt this option as a sensible general‐purpose compromise between robustness and simplicity, both of treatment and of interpretation of the outcome. Some theoretical arguments, outcomes of a few simulation experiments and various wide‐ranging examples with real data are provided in support of the claim.     

M5 competition uncertainty: Overdispersion,distributional forecasting,GAMLSS, and beyond

The M5 competition uncertainty track aims for probabilistic forecasting of sales of thousands of Walmart retail goods. We show that the M5 competition data face strong overdispersion and sporadic demand, especially zero demand. We discuss modeling issues concerning adequate probabilistic forecasting of such count data processes. Unfortunately, the majority of popular prediction methods used in the M5 competition (e.g. lightgbm and xgboost GBMs) fail to address the data characteristics, due to the considered objective functions. Distributional forecasting provides a suitable modeling approach to overcome those problems. The GAMLSS framework allows for flexible probabilistic forecasting using low-dimensional distributions. We illustrate how the GAMLSS approach can be applied to M5 competition data by modeling the location and scale parameters of various distributions, e.g. the negative binomial distribution. Finally, we discuss software packages for distributional modeling and their drawbacks, like the R package gamlss with its package extensions, and (deep) distributional forecasting libraries such as TensorFlow Probability.     

我国城市规模分布Pareto指数测算及影响因素分析

本文用OLS方法测算我国各省、三大区域以及全国的城市人口规模分布和经济规模分布Pareto指数(1997年、2000年和2003年),对Pareto指数进行跨区域和跨时间的对比分析,并实证分析我国城市规模分布的影响因素。分析表明,我国的城市规模分布显著地服从Pareto分布,并具有明显的结构性特征。工业化、产业结构以及运输能力对城市人口规模分布具有显著影响,而工业化和运输能力则是影响城市经济规模分布的重要因素。     

Akaike's Information Criterion,Cp and Estimators of Loss for Elliptically Symmetric Distributions

In this article, we develop a modern perspective on Akaike's information criterion and Mallows's C_p for model selection, and propose generalisations to spherically and elliptically symmetric distributions. Despite the differences in their respective motivation, C_p and Akaike's information criterion are equivalent in the special case of Gaussian linear regression. In this case, they are also equivalent to a third criterion, an unbiased estimator of the quadratic prediction loss, derived from loss estimation theory. We then show that the form of the unbiased estimator of the quadratic prediction loss under a Gaussian assumption still holds under a more general distributional assumption, the family of spherically symmetric distributions. One of the features of our results is that our criterion does not rely on the specificity of the distribution, but only on its spherical symmetry. The same kind of criterion can be derived for a family of elliptically contoured distribution, which allows correlations, when considering the invariant loss. More specifically, the unbiasedness property is relative to a distribution associated to the original density.     

Multiple Local Maxima in Restricted Likelihoods and Posterior Distributions for Mixed Linear Models

Scattered reports of multiple maxima in posterior distributions or likelihoods for mixed linear models appear throughout the literature. Less scrutinised is the restricted likelihood, which is the posterior distribution for a specific prior distribution. This paper surveys existing literature and proposes a unifying framework for understanding multiple maxima. For those problems with covariance structures that are diagonalisable in a specific sense, the restricted likelihood can be viewed as a generalised linear model with gamma errors, identity link and a prior distribution on the error variance. The generalised linear model portion of the restricted likelihood can be made to conflict with the portion of the restricted likelihood that functions like a prior distribution on the error variance, giving two local maxima in the restricted likelihood. Applying in addition an explicit conjugate prior distribution to variance parameters permits a second local maximum in the marginal posterior distribution even if the likelihood contribution has a single maximum. Moreover, reparameterisation from variance to precision can change the posterior modality; the converse also is true. Modellers should beware of these potential pitfalls when selecting prior distributions or using peak‐finding algorithms to estimate parameters.     

流通企业发展代理制对策研究

本文从分析代理制的含义及发展代理制的必要性出发,指出了目前我国代理制运行中存在的各种问题,从政府的宏观政策及企业的微观选择两方面提出相应的对策建议,以更好地推广代理制,实现物流合理化。     

Common inventory modeling assumptions that fall short: Arborescent networks, Poisson demand, and single-echelon approximations

Traditional multi-echelon inventory theory focuses on arborescent supply chains that use a central warehouse which replenishes remote warehouses. The remote warehouses serve customers in their respective regions. Common assumptions in the academic literature include use of the Poisson demand process and instantaneous unit-by-unit replenishment. In the practitioner literature, single-echelon approximations are advised for setting safety stock to deal with lead time, demand, and supply variations in these settings. Using data from a U.S. supplier of home improvement products, we find that neither the assumptions from the academic literature nor the approximations from the practitioner literature necessarily work well in practice.In a variation of the strictly arborescent supply chain, the central warehouse at our real company not only replenishes other warehouses but also meets demand from customers in the region near the central warehouse. In this paper, we study this dual-role central warehouse structure, which we believe is common in practice. Using high and low volume product demand data from this company, we use Monte Carlo simulations to study the impact of (1) the use of a dual-role centralized warehouse, (2) common demand assumptions made in multi-echelon research, and (3) single-echelon approximations for managing a multi-echelon supply chain. We explore each of these under both centralized and decentralized control logic. We find that the common assumptions of theoretical models impede their usefulness and that heuristics that ignore the actual supply chain structure fail to account for additional opportunities to utilize safety stock more effectively. Researchers should be aware of the gap between standard assumptions in traditional literature and actual practice, and critically evaluate their assumptions to find a reasonable balance between tractability and relevance.