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A Family of Goodness‐of‐Fit Tests for Copulas Based on Characteristic Functions
Author(s) -
Bahraoui Tarik,
Bouezmarni Taoufik,
Quessy JeanFrançois
Publication year - 2018
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12300
Subject(s) - mathematics , copula (linguistics) , goodness of fit , estimator , statistics , nonparametric statistics , parametric statistics , statistical hypothesis testing , econometrics
A general class of rank statistics based on the characteristic function is introduced for testing goodness‐of‐fit hypotheses about the copula of a continuous random vector. These statistics are defined as L 2 weighted functional distances between a nonparametric estimator and a semi‐parametric estimator of the characteristic function associated with a copula. It is shown that these statistics behave asymptotically as degenerate V ‐statistics of order four and that the limit distributions have representations in terms of weighted sums of independent chi‐square variables. The consistency of the tests against general alternatives is established and an asymptotically valid parametric bootstrap is suggested for the computation of the critical values of the tests. The behaviour of the new tests in small and moderate sample sizes is investigated with the help of simulations and compared with a competing test based on the empirical copula. Finally, the methodology is illustrated on a five‐dimensional data set.