Lorenz-generated bivariate Archimedean copulas

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

Mathematics ◽

10.3390/math8040625 ◽

2020 ◽

Vol 8 (4) ◽

pp. 625

Author(s):

Gheorghiță Zbăganu

Keyword(s):

Lorenz Curve ◽

Convergent Sequence ◽

Deterministic Case ◽

Asymptotic Results ◽

Lorenz Curves ◽

The Lorenz Curve

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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Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽

10.3390/math9090981 ◽

2021 ◽

Vol 9 (9) ◽

pp. 981

Author(s):

Patricia Ortega-Jiménez ◽

Miguel A. Sordo ◽

Alfonso Suárez-Llorens

Keyword(s):

Financial Risk ◽

Sufficient Conditions ◽

Random Variables ◽

Stochastic Ordering ◽

Random Variable ◽

Distribution Functions ◽

Stochastic Comparisons ◽

Stochastic Orderings ◽

Absolute Value ◽

The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

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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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Extremal behavior of Archimedean copulas

Advances in Applied Probability ◽

10.1239/aap/1300198519 ◽

2011 ◽

Vol 43 (1) ◽

pp. 195-216 ◽

Cited By ~ 20

Author(s):

Martin Larsson ◽

Johanna Nešlehová

Keyword(s):

Domain Of Attraction ◽

Tail Dependence ◽

Dependence Structure ◽

Tail Behavior ◽

Archimedean Copulas ◽

Radial Part ◽

Stochastic Representation ◽

Lower Tail ◽

Extremal Behavior ◽

Practical Implications

We show how the extremal behavior of d-variate Archimedean copulas can be deduced from their stochastic representation as the survival dependence structure of an ℓ1-symmetric distribution (see McNeil and Nešlehová (2009)). We show that the extremal behavior of the radial part of the representation is determined by its Williamson d-transform. This leads in turn to simple proofs and extensions of recent results characterizing the domain of attraction of Archimedean copulas, their upper and lower tail-dependence indices, as well as their associated threshold copulas. We outline some of the practical implications of their results for the construction of Archimedean models with specific tail behavior and give counterexamples of Archimedean copulas whose coefficient of lower tail dependence does not exist.

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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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