Abstract:We discuss the so-called "simplifying assumption" of conditional copulas in a general framework. We introduce several tests of the latter assumption for non-and semiparametric copula models. Some related test procedures based on conditioning subsets instead of point-wise events are proposed. The limiting distributions of such test statistics under the null are approximated by several bootstrap schemes, most of them being new. We prove the validity of a particular semiparametric bootstrap scheme. Some simulations illustrate the relevance of our results.

Extending the results of Bellec, Lecué and Tsybakov [1] to the setting of sparse highdimensional linear regression with unknown variance, we show that two estimators, the Square-Root Lasso and the Square-Root Slope can achieve the optimal minimax prediction rate, which is (s/n) log (p/s), up to some constant, under some mild conditions on the design matrix. Here, n is the sample size, p is the dimension and s is the sparsity parameter. We also prove optimality for the estimation error in the lq-norm, with q ∈ [1, 2] for the Square-Root Lasso, and in the l2 and sorted l1 norms for the Square-Root Slope. Both estimators are adaptive to the unknown variance of the noise. The Square-Root Slope is also adaptive to the sparsity s of the true parameter. Next, we prove that any estimator depending on s which attains the minimax rate admits an adaptive to s version still attaining the same rate.We apply this result to the Square-root Lasso. Moreover, for both estimators, we obtain valid rates for a wide range of confidence levels, and improved concentration properties as in [1] where the case of known variance is treated. Our results are non-asymptotic. MCS: Primary 62G08; secondary 62C20, 62G05.for any confidence level and for the risk in expectation. However, the estimators considered in [1-3] require the knowledge of the noise variance σ 2 . To our knowledge, no polynomial-time methods, which would be at the same time optimal in a minimax sense and adaptive both to σ and s are available in the literature.Estimators similar to the Lasso, but adaptive to σ are the Square-Root Lasso and the related Scaled Lasso, introduced by Sun and Zhang [13] and Belloni, Chernozhukov and Wang [4]. It has been shown to achieve the rate (s/n) log(p) in deviation with the value of the tuning parameter depending on the confidence level. A variant of this estimator is the Heteroscedastic Square-Root Lasso, which is studied in more general nonparametric and semiparametric setups by Belloni, Chernozhukov and Wang [5], but it also achieves the rate (s/n) log(p) and depends on the confidence level. We refer to the book by Giraud [8] for the link between the Lasso and the Square-Root Lasso and a short proof of oracle inequalities for the Square-root Lasso. In summary, there are two points to improve for the Square-root Lasso method:(i) The available results on oracle inequalities are valid only for the estimators depending on the confidence level. Thus, one cannot have an oracle inequality for one given estimator at any confidence level except the one that was used to design it.(ii) The obtained rate is (s/n) log(p) which is greater than the minimax rate (s/n) log(p/s).The Slope, which is an acronym for Sorted L-One Penalized Estimation, is an estimator introduced by Bogdan et al. [7], that is close to the Lasso, but uses the sorted l 1 norm instead of the standard l 1 norm for penalization. Su and Candès [12] proved that, as opposed to the Lasso, the Slope estimator is asymptotically minimax, in the sense that it attains the rate (s/n) log(p/...

We study nonparametric estimators of conditional Kendall's tau, a measure of concordance between two random variables given some covariates. We prove non-asymptotic pointwise and uniform bounds, that hold with high probabilities. We provide "direct proofs" of the consistency and the asymptotic law of conditional Kendall's tau. A simulation study evaluates the numerical performance of such nonparametric estimators.

We show how the problem of estimating conditional Kendall's tau can be rewritten as a classification task. Conditional Kendall's tau is a conditional dependence parameter that is a characteristic of a given pair of random variables. The goal is to predict whether the pair is concordant (value of 1) or discordant (value of −1) conditionally on some covariates. We prove the consistency and the asymptotic normality of a family of penalized approximate maximum likelihood estimators, including the equivalent of the logit and probit regressions in our framework. Then, we detail specific algorithms adapting usual machine learning techniques, including nearest neighbors, decision trees, random forests and neural networks, to the setting of the estimation of conditional Kendall's tau. Finite sample properties of these estimators and their sensitivities to each component of the data-generating process are assessed in a simulation study. Finally, we apply all these estimators to a dataset of European stock indices.