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On the weak convergence of the empirical conditional copula under a simplifying assumption | | François Portier
; Johan Segers
; | | Date: |
20 Nov 2015 | | Abstract: | When the copula of the conditional distribution of two random variables given
a covariate does not depend on the value of the covariate, two conflicting
intuitions arise about the best possible rate of convergence attainable by
nonparametric estimators of that copula. In the end, any such estimator must be
based on the marginal conditional distribution functions of the two dependent
variables given the covariate, and the best possible rates for estimating such
localized objects is slower than the parametric one. However, the invariance of
the conditional copula given the value of the covariate suggests the
possibility of parametric convergence rates. The more optimistic intuition is
shown to be correct, confirming a conjecture supported by extensive Monte Carlo
simulations by I. Hobaek Haff and J. Segers [Computational Statistics and Data
Analysis 84:1--13, 2015] and improving upon the nonparametric rate obtained
theoretically by I. Gijbels, M. Omelka and N. Veraverbeke [Scandinavian Journal
of Statistics 2015, to appear]. The novelty of the proposed approach lies in
the double smoothing procedure employed for the estimator of the marginal
cumulative distribution functions. Under mild conditions on the bandwidth
sequence, the estimator is shown to take values in a certain class of smooth
functions, the class having sufficiently small entropy for empirical process
arguments to work. The copula estimator itself is asymptotically
undistinguishable from a kind of oracle empirical copula, making it appear as
if the marginal conditional distribution functions were known. | | Source: | arXiv, 1511.6544 | | Services: | Forum | Review | PDF | Favorites | |
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