Markov morphisms: a combined copula and mass transportation approach to multivariate quantiles

• Autorzy: Faugeras O. P. , Rüschendorf L.

Identyfikatory

DOI

10.14708/ma.v45i1.2921

Warianty tytułu

PL

Morfizmy markowskie: zespolone podejście do wielowymiarowych kwantyli oparte o funkcje łącznikowe i zagadnienie transportowe

• Języki publikacji: EN

Abstrakty

EN

Our purpose is both conceptual and practical. On the one hand, we discuss the question which properties are basic ingredients of a general conceptual notion of a multivariate quantile. We propose and argue that the object “quantile” should be defined as a Markov morphism which carries over similar algebraic, ordering and topological properties as known for quantile functions on the real line. On the other hand, we also propose a practical quantile Markov morphism which combines a copula standardization and the recent optimal mass transportation method of Chernozhukov et al.(2017). Its empirical counterpart has the advantages of being a bandwidth-free, monotone invariant, a.s. consistent transformation. The proposed approach gives a general and unified framework to quantiles and their corresponding depth areas, for both a continuous or a discrete multivariate distribution.

PL

W artykule zaproponowano pewien sposobu wprowadzania kwantyli wielowymiarowych, jak i metody ich wyznaczania. Z jednej strony podstawą rozważań są podstawowe własności uogólnionego pojęcia wielowymiarowego kwantyla, który jest morfizmem markowskim, zachowującym podobne własności algebraiczne, topologiczne oraz porządku, jakie znamy dla linii kwantylowych na prostej rzeczywistej. Z drugiej zaś strony, zaproponowano morfiz markowski, który łączy standaryzowaną kopułę (funkcję łącznikową) z zastosowaniem zagadnienia transportowego (v. Chernozhukov et al.(2017). Proponowane podejście daje ogólne i jednolite podejście do definicji kwantyli i ich estymacji, zarówno dla ciągłych, jak i dyskretnych rozkładów wielowymiarowych.

Słowa kluczowe

EN

statistical depth; vector quantiles; Markov morphism; copula; mass transportation

PL

kwantyle wielowymiarowe; kopuły; zależność statystyczna; morfizmy markowskie

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Mathematica Applicanda
• Rocznik: 2017
• Tom: Vol. 45, No. 1
• Strony: 21--63
• Opis fizyczny: Bibliogr. 64 poz., fot.

Twórcy

autor

Faugeras O. P.

• University of Toulouse Capitole, Toulouse School of Economics, Manufacture des Tabacs, Bureau MF319, 21 Allée de Brienne, 31000 Toulouse, France

autor

Rüschendorf L.

• University of Freiburg, Abteilung für Mathematische Stochastik, Albert-Ludwigs University of Freiburg, Eckerstraße 1, D–79104 Freiburg, Germany

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Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).