The Pareto optimality distribution

Pareto optimality (sometimes known as Pareto efficiency) is an important notion in social sciences and related areas, see e.g. Klaus (2006), Chun (2005), Hild (2004), Kibris (2003), Nandeibam (2003), Papai (2001), Peris and Sanchez (2001), Brams and Fishburn (2000), Denicolo (1999), Klaus et al. (1998), Peremans et al. (1997), and Vohra (1992). This notion invariably involves the comparison of the utility of one outcome versus another, i.e. the ratio of two utilities or in general the ratio of two random variables. In this note, we derive the exact distribution of the ratio X/(X + Y) when X and Y are Pareto random variables, Pareto distribution being the first and the most popular distribution used in social sciences and related areas.     

Multivariate survival functions characterized by constant product of mean remaining lives and hazard rates

Under the assumption that the products of multivariate mean remaining lives and hazard rates are the same constant, it is shown that the corresponding multivariate survival function belongs to one of three families: (1) multivariate Gumbel exponential distribution; (2) multivariate Lomax (Pareto type II) distribution; (3) multivariate rescaled Dirichlet distribution. This result is then used to derive another characterization of the latter two families based on the residual life distribution.     

The size distribution of cities: An examination of the Pareto law and primacy

This paper examines the Pareto and primacy measures of the size distribution of cities. The mean Pareto exponent for a sample of 44 countries is 1.136, somewhat greater than the exponent of one implied by the rank-size rule. We find that value of the Pareto exponent is quite sensitive to the definition of the city and the choice of city sample size. The significance of non-linear terms in variants of the Pareto distribution also indicate that the rank-size rule is only a first approximation to a complete characterization of the size distribution of cities within a country. The relatively low correlation between primacy and Pareto measures confirms the need for a variety of measures of city size distributions. This paper also suggests that large cities are growing faster than small cities in most of the countries in our sample. This is indicated by the positive coefficient on the first non-linear term introduced into the Pareto equation. Finally, variations in the Pareto exponent and measures of primacy are partly explained by economic, demographic, and geographic factors.     

Record values of the Lomax distribution

In this paper, firstly an introduction to the idea of record values for a sequence of independent and identically (Lomax or Pareto II) distributed random variables is given. Some of the distributional properties of these record values and moments up to second order are derived. These moments clearly depend on the location, scale and shape parameter of the Lomax distribution and two types of estlmators of these parameters, based on a series of observed record values are presented.     

A product Pareto distribution

The Pareto distributions are becoming increasing prominent in several applied areas. In this note, a new Pareto distribution is introduced. It takes the form of the product of two Pareto probability density functions. Various structural properties of this distribution are derived, including its cumulative distribution function, moments, mean deviation about the mean, mean deviation about the median, entropy, asymptotic distribution of the extreme order statistics, method of moments estimates, maximum likelihood estimates and the Fisher information matrix. The calculations involve the use of several special functions.     

On the distribution of city sizes

The city size distribution in many countries is remarkably well described by a Pareto distribution. We derive conditions that standard urban models must satisfy in order to explain this regularity. We show that under general conditions urban models must have (i) a balanced growth path and (ii) a Pareto distribution for the underlying source of randomness. In particular, one of the following combinations can induce a Pareto distribution of city sizes: (i) preferences for different goods follow reflected random walks, and the elasticity of substitution between goods is 1; or (ii) total factor productivities of different goods follow reflected random walks, and increasing returns are equal across goods.     

Joint confidence region for the parameters of pareto distribution

The Pareto distribution is commonly used by economists as a model for the distribution of incomes. Separate confidence intervals or approximate confidence intervals for the parameters of Pareto distribution were discussed by some authors. This paper discusses exact joint confidence region for the parameters of Pareto distribution. The method can be used for both complete samples and type II censored samples.     

A residual inaccuracy measure based on the relevation transform

Inaccuracy and information measures based on the cumulative residual entropy are useful in various fields, and are attracting increasing attention in Probability Theory and Statistics. In this paper, we introduce and study an inaccuracy measure concerning the relevation transform of two nonnegative continuous random variables. We investigate various distributional properties and characterization results that are based on the mean residual lifetime and involve the generalized Pareto distribution. A connection with the proportional hazards model is also provided. We obtain comparison results involving the proposed inaccuracy measure and some existing inaccuracy measures. Some illustrative examples are finally given.     

Distribution of product and quotient of Pareto variates

Density and distribution functions are obtained for the product ofn independent Pareto variates with different parameters as well as for the quotient of two independent Pareto variates.Results are given in such a form that for several appropriate values of the parameters, both density and distribution functions can be tabulated.     

Approximate confidence and tolerance limits for the discrete Pareto distribution for characterizing extremes in count data

Statistical tolerance intervals for discrete distributions are widely employed for assessing the magnitude of discrete characteristics of interest in applications like quality control, environmental monitoring, and the validation of medical devices. For such data problems, characterizing extreme counts or outliers is also of considerable interest. These applications typically use traditional discrete distributions, like the Poisson, binomial, and negative binomial. The discrete Pareto distribution is an alternative yet flexible model for count data that are heavily right‐skewed. Our contribution is the development of statistical tolerance limits for the discrete Pareto distribution as a strategy for characterizing the extremeness of observed counts in the tail. We discuss the coverage probabilities of our procedure in the broader context of known coverage issues for statistical intervals for discrete distributions. We address this issue by applying a bootstrap calibration to the confidence level of the asymptotic confidence interval for the discrete Pareto distribution's parameter. We illustrate our procedure on a dataset involving cyst formation in mice kidneys.     

US city-size distribution and space

ABSTRACT

This paper focuses on spatial city-size distribution in the United States. It proposes a new distance-based approach to analyze the influence of distance on the city-size distribution parameter by considering the Pareto distribution and using data from different definitions of US cities in 2010. Considering all possible combinations of cities within a 300-mile radius, the results indicate that the Pareto distribution cannot be rejected in most cases regardless of city size. Placebo regressions validate the results, thereby confirming the significant effect of geography on the Pareto exponent.     

Moment inequalities for order statistics with applications to characterizations of distributions

General inequalities of Hölder type between moments of order statistics and moments of record values respectively are derived. Special choices of the involved sample sizes and ranks and discussions of when equality is attained in these inequalities yield several characterizations of well known distributions, such as the uniform, polynomial, Pareto, reflected Pareto, exponential, Weibull distribution and some others.     

THE USE OF THE LORENZ CURVE AND THE PARETO DISTRIBUTION IN INTERNAL PAY STRUCTURING: A RESEARCH NOTE

This note presents some findings from the early stages of research into the effectiveness of company pay structures. ¹ 1 A two-year project sponsored by the S.S.R.C. The authors gratefully acknowledge the assistance of the S.S.R.C. in this work.
The use of the Lorenz curve as a measure of inequality of distribution of a firm's payroll is described. Values for L, the Lorenz coefficient, are presented for the total payroll of a chemical processing firm and for the earnings of manual workers in a medium-heavy engineering company. The use of the Pareto distribution for management salary structuring is demonstrated. Values for α, the Pareto coefficient, are presented for two chemical companies and two engineering firms. Some possible practical uses for such Lorenz and Pareto values are suggested.     

Estimation of econometric models of some discrete games

This paper discusses, estimates and compares some microeconometric models for simultaneous discrete endogenous variables. The models are based on the assumption that observed endogenous variables represent the outcome of a static discrete game. I discuss models based on non-cooperative equilibrium concepts (Nash, Stackelberg), as well as models which presume Pareto optimality of observed outcomes. The models are estimated using data on the joint labor force participation decisions of husbands and wives in a sample of Dutch households.     

Sociological Models Based on Fréchet Random Variables

Products of random variables are of both practical and theoretical significance to social scientists. This has increased the need to have available the widest possible range of statistical results on products of random variables. In this note, the distribution of the product XY is derived when X and Y are independent Fréchet random variables. Extensive tabulations of the associated percentage points are also given.     

An agent-based model of the observed distribution of wealth in the United States

Pareto cautiously asserted that the wealth and income distributions which bear his name are universal, basing his argument on observations of this distribution in many different types of economies. In this paper, we present an agent based model (and a scalable approximation of it) in a closely related spirit. The central feature of this model is that wealth enables an individual to secure more wealth. Specifically, the important and novel feature of this model is its ability to simultaneously produce both the Pareto distribution observed in empirical data for the top 10% of the population and the exponential distribution observed for the lower 90%. We show that the model produces these distributions of wealth when initialized with an equitable distribution. Then, using historical data, we initialize the model with US wealth shares in 1988 and show that the model tracks wealth share changes from 1988 to 2012. Simulations to 2088 project that the top 0.01% of the population will possess more than 70% of the total wealth in the economy.     

The size distribution of Chinese cities

This paper uses urban data to investigate two important issues regarding city sizes in China, the relative growth of cities and the nature of the city size distribution. The manner in which cities of different sizes grow relative to each other is examined and, contrary to the common empirical finding that the relative size and rank of cities remains stable over time, it is found that the Economic Reforms and the One Child Policy since 1979 have delivered significant structural change in the Chinese urban system. The city size distribution remains stable before the reforms but exhibits a convergent growth pattern in the post-reform period. The theoretical literature on city sizes highlights a link between log normal and Pareto distributions for city sizes prompting the employment of Pearson goodness-of-fit tests to examine directly which theoretical distribution provides the best approximation to the empirical city size distribution. Contrary to the evidence for other countries, a log normal rather than Pareto specification turns out to be the preferred distribution.     

Cost sharing: efficiency and implementation

We study environments where a production process is jointly shared by a finite group of agents. The social decision involves the determination of input contribution and output distribution. We define a competitive solution when there is decreasing-returns-to-scale which leads to a Pareto optimal outcome. Since there is a finite number of agents, the competitive solution is prone to manipulation. We construct a mechanism for which the set of Nash equilibria coincides with the set of competitive solution outcomes. We define a marginal-cost-pricing equilibrium (MCPE) solution for environments with increasing returns to scale. These solutions are Pareto optimal under certain conditions. We construct another mechanism that realizes the MCPE.     

Bivariate semi-Pareto minification processes

Marshall-Olkin bivariate semi-Pareto distribution (MO-BSP) and Marshall-Olkin bivariate Pareto distribution (MO-BP) are introduced and studied. AR(1) and AR(k) time series models are developed with minification structure having MO-BSP stationary marginal distribution. Various characterizations are investigated.Acknowledgements. The authors thank the Editor and the referee for their valuable suggestions which led to an improved version of the original paper. The first author is grateful to the University Grants Commission of India for the support under Teacher Fellowship Scheme.     

Estimation in the pareto distribution

The unique minimum variance unbiased (UMVU) estimate of the probability distribution function of the Pareto distribution is derived. It is shown that the distribution function and ther ^th moment associated with the UMVU estimate are also UMVU estimators. The p.d.f. and its estimator are compared graphically. An estimate of the 100p ^th percentile is given. It is seen that a function of this estimator has a chi-square distribution.