The Gini coefficient and discontinuity
This article reveals a discontinuity in the mapping from a Lorenz curve to the associated cumulative distribution function. The problem is of a mathematical nature—based on an analysis of the transformation between the distribution function of a bound random variable and its Lorenz curve. It will be...
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| Format: | Article |
| Language: | English |
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Taylor & Francis Group
2022-12-01
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| Series: | Cogent Economics & Finance |
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| Online Access: | https://www.tandfonline.com/doi/10.1080/23322039.2022.2072451 |
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| _version_ | 1850185293439696896 |
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| author | Jens Peter Kristensen |
| author_facet | Jens Peter Kristensen |
| author_sort | Jens Peter Kristensen |
| collection | DOAJ |
| description | This article reveals a discontinuity in the mapping from a Lorenz curve to the associated cumulative distribution function. The problem is of a mathematical nature—based on an analysis of the transformation between the distribution function of a bound random variable and its Lorenz curve. It will be proven that the transformation from a normalized income distribution to its Lorenz curve is a continuous bijection with respect to the [Formula: see text] ([0,1])-metric—for every q ≥ 1. The inverse transformation, however, is not continuous for any q ≥ 1. This implies a more careful attitude when interpreting the value of a Gini coefficient. A further problem is that if you have estimated a Lorenz curve from empirical data,then you cannot trust that the associated distribution is a good estimate of the true income distribution. |
| format | Article |
| id | doaj-art-e2cf4541bc624f4a9bab6f78cdb4e814 |
| institution | OA Journals |
| issn | 2332-2039 |
| language | English |
| publishDate | 2022-12-01 |
| publisher | Taylor & Francis Group |
| record_format | Article |
| series | Cogent Economics & Finance |
| spelling | doaj-art-e2cf4541bc624f4a9bab6f78cdb4e8142025-08-20T02:16:46ZengTaylor & Francis GroupCogent Economics & Finance2332-20392022-12-0110110.1080/23322039.2022.2072451The Gini coefficient and discontinuityJens Peter Kristensen0Retired lecturer, M. Sc. in mathematics, DenmarkThis article reveals a discontinuity in the mapping from a Lorenz curve to the associated cumulative distribution function. The problem is of a mathematical nature—based on an analysis of the transformation between the distribution function of a bound random variable and its Lorenz curve. It will be proven that the transformation from a normalized income distribution to its Lorenz curve is a continuous bijection with respect to the [Formula: see text] ([0,1])-metric—for every q ≥ 1. The inverse transformation, however, is not continuous for any q ≥ 1. This implies a more careful attitude when interpreting the value of a Gini coefficient. A further problem is that if you have estimated a Lorenz curve from empirical data,then you cannot trust that the associated distribution is a good estimate of the true income distribution.https://www.tandfonline.com/doi/10.1080/23322039.2022.2072451inequalityprobabilitymath analysisdiscontinuity |
| spellingShingle | Jens Peter Kristensen The Gini coefficient and discontinuity Cogent Economics & Finance inequality probability math analysis discontinuity |
| title | The Gini coefficient and discontinuity |
| title_full | The Gini coefficient and discontinuity |
| title_fullStr | The Gini coefficient and discontinuity |
| title_full_unstemmed | The Gini coefficient and discontinuity |
| title_short | The Gini coefficient and discontinuity |
| title_sort | gini coefficient and discontinuity |
| topic | inequality probability math analysis discontinuity |
| url | https://www.tandfonline.com/doi/10.1080/23322039.2022.2072451 |
| work_keys_str_mv | AT jenspeterkristensen theginicoefficientanddiscontinuity AT jenspeterkristensen ginicoefficientanddiscontinuity |