Positive quadrant dependence tests for copulas

Abstract

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a “distance” between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov–Smirnov, Cramér–von‐Mises, and Anderson–Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555–581; 2010 © 2010 Statistical Society of Canada

A new family of copula-based concordance orderings of random pairs: Properties and nonparametric tests

“…(X 1 , X 2 ) s− o (X ⊥ 1 , X ⊥ 2 )), where (X ⊥ 1 , X ⊥ 2 ) is a copy of (X 1 , X 2 ), but with independent components. The statistical tools of Section 4 and Section 5 could then be adapted in order to provide not only interesting extensions of the tests of Scaillet (2005) and Gijbels, Omelka and Sznajder (2010) to s-PQD and s-NQD, Now since (X 1 , X 2 ) s− o (Y 1 , Y 2 ) holds if and only if (F 1 (X 1 ),F 2 (X 2 )) s−ICX (Ḡ 1 (Y 1 ),Ḡ 2 (Y 2 )), Characterization 3.1 of Denuit, Lefèvre and Mesfioui (1999) for the s-increasing convex order may be formulated as E{O s (u 1 , u 2 ;F 1 (X 1 ),F 2 (X 2 ))} ≤ E{O s (u 1 , u 2 ;Ḡ 1 (Y 1 ),Ḡ 2 (Y 2 ))}, E{O (i1,s2) (0, u 2 ;F 1 (X 1 ),F 2 (X 2 ))}≤E{O (i1,s2) (0, u 2 ;Ḡ 1 (Y 1 ),Ḡ 2 (Y 2 ))} ∀i 1 <s 1 , E{O (s1,i2) (u 1 , 0;F 1 (X 1 ),F 2 (X 2 ))}≤E{O (s1,i2) (u 1 , 0;Ḡ 1 (Y 1 ),Ḡ 2 (Y 2 ))} ∀i 2 <s 2 , E{O i (0, 0;F 1 (X 1 ),F 2 (X 2 ))} ≤ E{O i (0, 0;Ḡ 1 (Y 1 ),Ḡ 2 (Y 2 ))} ∀i < s.…”

Nonparametric tests for tail monotonicity

“…While testing for the latter hypothesis, i.e., that Y is positively regression dependent on X, or testing for positive quadrant dependence has recently drawn some attention in the literature, see Delgado and Escanciano (2012); Lee et al (2009); Denuit and Scaillet (2004); Scaillet (2005); Gijbels et al (2010), there does not exist an omnibus test for the hypothesis of any of the tail monotonicity concepts defined above, at least to the best of our knowledge. It is the purpose of this paper to fill this gap.…”

“…data sets (e.g. Delgado and Escanciano (2012);Scaillet (2005); Gijbels et al (2010)). Exploiting a recent block multiplier bootstrap method, see Bücher and Ruppert (2012), our proposed test on tail monotonicity goes beyond this and can also be applied to serially dependent, strongly mixing data sets.…”

“…In the context of testing for the similar hypothesis of quadrant dependence, Gijbels et al (2010) proposed to approximate critical values of their corresponding test statistic by simulating from the least favorable copula under H 0 . In our context the least favorable copula, i.e., the copula which is closest to the alternative, is that of a constant function u → C(u, v)/u for all v ∈ [0, 1].…”

Confidence band for expectation dependence with applications

“…In order to study on stronger definitions of dependence, Lehmann (1966) makes far-reaching contributions to the characterization of quadrant dependence. For literatures about this concept, see for instance, Denuit and Scaillet (2004), Scaillet (2005), Kallenberg (2008), Dhaene et al (2009), Gijbels et al (2010) and Ledwina and Wylupek (2014).…”