Asymptotic Results in Broken Stick Models: The Approach via Lorenz Curves

A stick of length 1 is broken at random into n smaller sticks. How much inequality does this procedure produce? What happens if, instead of breaking a stick, we break a square? What happens asymptotically? Which is the most egalitarian distribution of the smaller sticks (or rectangles)? Usually, when studying inequality, one uses a Lorenz curve. The more egalitarian a distribution, the closer the Lorenz curve is to the first diagonal of [ 0 , 1 ] 2 . This is why in the first section we study the space of Lorenz curves. What is the limit of a convergent sequence of Lorenz curves? We try to answer these questions, firstly, in the deterministic case and based on the results obtained there in the stochastic one.

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Convergence theorems for empirical Lorenz curves and their inverses

Advances in Applied Probability ◽

10.2307/1426700 ◽

1977 ◽

Vol 9 (4) ◽

pp. 765-791 ◽

Cited By ~ 68

Author(s):

Charles M. Goldie

Keyword(s):

Lorenz Curve ◽

Population Distribution ◽

Finite Variance ◽

Cumulative Proportion ◽

Lorenz Curves ◽

Distribution Of Wealth ◽

Empirical Distributions ◽

The Lorenz Curve ◽

Definition Of ◽

Functional Central Limit Theorems

The Lorenz curve of the distribution of ‘wealth’ is a graph of cumulative proportion of total ‘wealth’ owned, against cumulative proportion of the population owning it. This paper uses Gastwirth's definition of the Lorenz curve which applies to a general probability distribution on (0, ∞) having finite mean; thus it applies both to a ‘population’ distribution and to empirical distributions obtained on sampling. The Lorenz curves of the latter are proved to converge, with probability 1, uniformly to the former, and similarly for their inverses. Modified Lorenz curves are also defined, which treat atoms of differently, and these and their inverses are proved strongly consistent. Functional central limit theorems are then proved for empirical Lorenz curves and their inverses, under condition that be continuous and have finite variance. A mild variation condition is also needed in some circumstances. If the support of is connected, the weak convergence is relative to C[0, 1] with uniform topology, otherwise to D[0, 1] with M1 topology. Selected applications are discussed, one being to the Gini coefficient.

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Convergence theorems for empirical Lorenz curves and their inverses

Advances in Applied Probability ◽

10.1017/s0001867800029177 ◽

1977 ◽

Vol 9 (04) ◽

pp. 765-791 ◽

Cited By ~ 15

Author(s):

Charles M. Goldie

Keyword(s):

Lorenz Curve ◽

Population Distribution ◽

Finite Variance ◽

Cumulative Proportion ◽

Lorenz Curves ◽

Distribution Of Wealth ◽

Empirical Distributions ◽

The Lorenz Curve ◽

Definition Of ◽

Functional Central Limit Theorems

The Lorenz curve of the distribution of ‘wealth’ is a graph of cumulative proportion of total ‘wealth’ owned, against cumulative proportion of the population owning it. This paper uses Gastwirth's definition of the Lorenz curve which applies to a general probability distribution on (0, ∞) having finite mean; thus it applies both to a ‘population’ distributionand to empirical distributions obtained on sampling. The Lorenz curves of the latter are proved to converge, with probability 1, uniformly to the former, and similarly for their inverses. Modified Lorenz curves are also defined, which treat atoms ofdifferently, and these and their inverses are proved strongly consistent. Functional central limit theorems are then proved for empirical Lorenz curves and their inverses, under condition thatbe continuous and have finite variance. A mild variation condition is also needed in some circumstances. If the support ofis connected, the weak convergence is relative toC[0, 1] with uniform topology, otherwise toD[0, 1] withM1topology. Selected applications are discussed, one being to the Gini coefficient.

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Lorenz-generated bivariate Archimedean copulas

Dependence Modeling ◽

10.1515/demo-2020-0011 ◽

2020 ◽

Vol 8 (1) ◽

pp. 186-209

Author(s):

Andrea Fontanari ◽

Pasquale Cirillo ◽

Cornelis W. Oosterlee

Keyword(s):

Lorenz Curve ◽

Tail Dependence ◽

Stochastic Ordering ◽

Random Variable ◽

Wealth Inequality ◽

Singular Part ◽

Integral Transforms ◽

Archimedean Copulas ◽

Lorenz Curves ◽

The Lorenz Curve

AbstractA novel generating mechanism for non-strict bivariate Archimedean copulas via the Lorenz curve of a non-negative random variable is proposed. Lorenz curves have been extensively studied in economics and statistics to characterize wealth inequality and tail risk. In this paper, these curves are seen as integral transforms generating increasing convex functions in the unit square. Many of the properties of these “Lorenz copulas”, from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating Lorenz curve. For instance, by selecting random variables with a lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An “alchemy” of Lorenz curves that can be used as general framework to build multiparametric families of copulas is also discussed.

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An empirical comparison of alternative functional forms for the Lorenz curve

Applied Economics Letters ◽

10.1080/13504850110054058 ◽

2002 ◽

Vol 9 (3) ◽

pp. 171-176 ◽

Cited By ~ 11

Author(s):

Kwang Soo Cheong

Keyword(s):

Lorenz Curve ◽

Empirical Comparison ◽

Functional Forms ◽

The Lorenz Curve

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The Contest for Olympic Success as a Public Good

Journal of Income Distribution ◽

10.25071/1874-6322.34371 ◽

2013 ◽

Author(s):

Loek Groot

Keyword(s):

Lorenz Curve ◽

Gini Index ◽

World Population ◽

Level Playing Field ◽

Actual Distribution ◽

Playing Field ◽

Inequality Measures ◽

The Rich ◽

The Lorenz Curve ◽

Rich Countries

In this study it is demonstrated that standard income inequality measures, such as the Lorenz curve and the Gini index, can successfully be applied to the distribution of Olympic success. Olympic success is distributed very unevenly, with the rich countries capturing a disproportionately higher share compared to their world population share, which suggests that the Olympic Games do not provide a level playing field. The actual distribution of Olympic success is compared with alternative hypothetical distributions, among which are chosen the distribution according to population shares, the welfare optimal distribution under the assumption of zero government expenditures, and the non-cooperating Nash-Cournot distribution. By way of conclusion, a device is proposed to make the distribution of Olympic success more equitable.

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