On a class of norms generated by nonnegative integrable distributions
We show that any distribution function on ℝd with nonnegative, nonzero and integrable marginal distributions can be characterized by a norm on ℝd+1, called F-norm. We characterize the set of F-norms and prove that pointwise convergence of a sequence of F-norms to an F-norm is equivalent to convergen...
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De Gruyter
2019-07-01
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| Series: | Dependence Modeling |
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| Online Access: | https://doi.org/10.1515/demo-2019-0014 |
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doaj-a003cf5277b14ddb9891027ada53314e2021-10-02T19:16:17ZengDe GruyterDependence Modeling2300-22982019-07-017125927810.1515/demo-2019-0014demo-2019-0014On a class of norms generated by nonnegative integrable distributionsFalk Michael0Stupfler Gilles1Institute of Mathematics, University of Würzburg, Würzburg, GermanySchool of Mathematical Sciences, University of Nottingham, Nottingham, UKWe show that any distribution function on ℝd with nonnegative, nonzero and integrable marginal distributions can be characterized by a norm on ℝd+1, called F-norm. We characterize the set of F-norms and prove that pointwise convergence of a sequence of F-norms to an F-norm is equivalent to convergence of the pertaining distribution functions in the Wasserstein metric. On the statistical side, an F-norm can easily be estimated by an empirical F-norm, whose consistency and weak convergence we establish.https://doi.org/10.1515/demo-2019-0014characteristic functiond-normempirical distribution functionhausdorff metricmultivariate distributionnormwasserstein metric60e1060g9962h0562h12 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Falk Michael Stupfler Gilles |
| spellingShingle |
Falk Michael Stupfler Gilles On a class of norms generated by nonnegative integrable distributions Dependence Modeling characteristic function d-norm empirical distribution function hausdorff metric multivariate distribution norm wasserstein metric 60e10 60g99 62h05 62h12 |
| author_facet |
Falk Michael Stupfler Gilles |
| author_sort |
Falk Michael |
| title |
On a class of norms generated by nonnegative integrable distributions |
| title_short |
On a class of norms generated by nonnegative integrable distributions |
| title_full |
On a class of norms generated by nonnegative integrable distributions |
| title_fullStr |
On a class of norms generated by nonnegative integrable distributions |
| title_full_unstemmed |
On a class of norms generated by nonnegative integrable distributions |
| title_sort |
on a class of norms generated by nonnegative integrable distributions |
| publisher |
De Gruyter |
| series |
Dependence Modeling |
| issn |
2300-2298 |
| publishDate |
2019-07-01 |
| description |
We show that any distribution function on ℝd with nonnegative, nonzero and integrable marginal distributions can be characterized by a norm on ℝd+1, called F-norm. We characterize the set of F-norms and prove that pointwise convergence of a sequence of F-norms to an F-norm is equivalent to convergence of the pertaining distribution functions in the Wasserstein metric. On the statistical side, an F-norm can easily be estimated by an empirical F-norm, whose consistency and weak convergence we establish. |
| topic |
characteristic function d-norm empirical distribution function hausdorff metric multivariate distribution norm wasserstein metric 60e10 60g99 62h05 62h12 |
| url |
https://doi.org/10.1515/demo-2019-0014 |
| work_keys_str_mv |
AT falkmichael onaclassofnormsgeneratedbynonnegativeintegrabledistributions AT stupflergilles onaclassofnormsgeneratedbynonnegativeintegrabledistributions |
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1716847529282240512 |