The geometry of the phosphodiester backbone was analyzed for 7739 dinucleotides from 447 selected crystal structures of naked and complexed DNA. Ten torsion angles of a near-dinucleotide unit have been studied by combining Fourier averaging and clustering. Besides the known variants of the A-, B- and Z-DNA forms, we have also identified combined A + B backbone-deformed conformers, e.g. with α/γ switches, and a few conformers with a syn orientation of bases occurring e.g. in G-quadruplex structures. A plethora of A- and B-like conformers show a close relationship between the A- and B-form double helices. A comparison of the populations of the conformers occurring in naked and complexed DNA has revealed a significant broadening of the DNA conformational space in the complexes, but the conformers still remain within the limits defined by the A- and B- forms. Possible sequence preferences, important for sequence-dependent recognition, have been assessed for the main A and B conformers by means of statistical goodness-of-fit tests. The structural properties of the backbone in quadruplexes, junctions and histone-core particles are discussed in further detail.

We reconsider the existing kernel estimators for a copula function, as proposed in Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445-464], Fermanian, Radulovič and Wegkamp [Bernoulli 10 (2004) 847-860] and Chen and Huang [Canad. J. Statist. 35 (2007) 265-282].All of these estimators have as a drawback that they can suffer from a corner bias problem. A way to deal with this is to impose rather stringent conditions on the copula, outruling as such many classical families of copulas. In this paper, we propose improved estimators that take care of the typical corner bias problem. For Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445-464] and Chen and Huang [Canad. J. Statist. 35 (2007) 265-282], the improvement involves shrinking the bandwidth with an appropriate functional factor; for Fermanian, Radulovič and Wegkamp [Bernoulli 10 (2004) 847-860], this is done by using a transformation. The theoretical contribution of the paper is a weak convergence result for the three improved estimators under conditions that are met for most copula families. We also discuss the choice of bandwidth parameters, theoretically and practically, and illustrate the finite-sample behaviour of the estimators in a simulation study. The improved estimators are applied to goodness-of-fit testing for copulas.

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Abstarct.  This paper is concerned with studying the dependence structure between two random variables Y1 and Y2 conditionally upon a covariate X. The dependence structure is modelled via a copula function, which depends on the given value of the covariate in a general way. Gijbels et al. (Comput. Statist. Data Anal., 55, 2011, 1919) suggested two non‐parametric estimators of the ‘conditional’ copula and investigated their numerical performances. In this paper we establish the asymptotic properties of the proposed estimators as well as conditional association measures derived from them. Practical recommendations for their use are then discussed.

This paper is concerned with studying the dependence structure between two random variables Y 1 and Y 2 in the presence of a covariate X , which affects both marginal distributions but not the dependence structure. This is reflected in the property that the conditional copula of Y 1 and Y 2 given X , does not depend on the value of X . This latter independence often appears as a simplifying assumption in pair-copula constructions. We introduce a general estimator for the copula in this specific setting and establish its consistency. Moreover, we consider some special cases, such as parametric or nonparametric location-scale models for the effect of the covariate X on the marginals of Y 1 and Y 2 and show that in these cases, weak convergence of the estimator, at p n-rate, holds. The theoretical results are illustrated by simulations and a real data example.