2015/11/20 by François Portier, Johan Segers, Portier, François +1
Computer Science · Mathematics · #62G20 #62G30 #Bayesian Methods and Mixture Models #FOS: Mathematics #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1511.06544
openalex publication_date 2015/11/20 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
When the copula of the conditional distribution of two random variables given\na covariate does not depend on the value of the covariate, two conflicting\nintuitions arise about the best possible rate of convergence attainable by\nnonparametric estimators of that copula. In the end, any such estimator must be\nbased on the marginal conditional distribution functions of the two dependent\nvariables given the covariate, and the best possible rates for estimating such\nlocalized objects is slower than the parametric one. However, the invariance of\nthe conditional copula given the value of the covariate suggests the\npossibility of parametric convergence rates. The more optimistic intuition is\nshown to be correct, confirming a conjecture supported by extensive Monte Carlo\nsimulations by I. Hobaek Haff and J. Segers [Computational Statistics and Data\nAnalysis 84:1--13, 2015] and improving upon the nonparametric rate obtained\ntheoretically by I. Gijbels, M. Omelka and N. Veraverbeke [Scandinavian Journal\nof Statistics 2015, to appear]. The novelty of the proposed approach lies in\nthe double smoothing procedure employed for the estimator of the marginal\ncumulative distribution functions. Under mild conditions on the bandwidth\nsequence, the estimator is shown to take values in a certain class of smooth\nfunctions, the class having sufficiently small entropy for empirical process\narguments to work. The copula estimator itself is asymptotically\nundistinguishable from a kind of oracle empirical copula, making it appear as\nif the marginal conditional distribution functions were known.\n