Alexis Derumigny

Alberto Garzás-Villar, Alba Riera-Cardona, Alexis Derumigny et al.

Robotic haptic devices combined with virtual reality offer novel opportunities to train fine force generation, an essential yet overlooked component of post-stroke rehabilitation. This study proposes that manipulating the rendered dynamics of tangible virtual objects can be leveraged to train precise force control while engaging the somatosensory system. We conducted an experiment with fifty healthy participants who performed a curling-inspired task in which they had to stretch a virtual spring to generate a target release force to propel the stone to a predefined location on the ice sheet. During training, the spring's force-elongation relationship was modeled as either a linear or non-linear function, i.e., a Gaussian or antisymmetric Gaussian (AS-Gaussian) function with zero derivative at the release target force. Results indicate that the AS-Gaussian group consistently achieved higher force accuracy during training than the linear group, while the Gaussian group only outperformed the linear group toward the end of training. Analysis of personality traits revealed that higher Free Spirit scores were associated with poorer performance and reduced task exploration under Gaussian dynamics, whereas higher Transform-of-Challenge scores correlated with increased exploration. Despite these training effects, no significant differences in long-term retention were found across spring types or personality traits. Participants primarily relied on learned target elongation rather than target force, as evidenced by performance in a transfer task with a different stiffness but the same target force. While promising for somatosensory neurorehabilitation, these methods require refinement to reduce reliance on proprioceptive cues before testing with neurological patients.

Alexis Derumigny, Jean-David Fermanian, Aleksey Min

Several procedures have been recently proposed to test the simplifying assumption for conditional copulas. Instead of considering pointwise conditioning events, we study the constancy of the conditional dependence structure when some covariates belong to general borelian conditioning subsets. Several test statistics based on the equality of conditional Kendall's tau are introduced, and we derive their asymptotic distributions under the null. When such conditioning events are not fixed ex ante, we propose a data-driven procedure to recursively build such relevant subsets. It is based on decision trees that maximize the differences between the conditional Kendall's taus corresponding to the leaves of the trees. The performances of such tests are illustrated in a simulation experiment. Moreover, a study of the conditional dependence between financial stock returns is managed, given some clustering of their past values. The last application deals with the conditional dependence between coverage amounts in an insurance dataset.

Alexis Derumigny, Johannes Schmidt-Hieber

It is a common phenomenon that for high-dimensional and nonparametric statistical models, rate-optimal estimators balance squared bias and variance. Although this balancing is widely observed, little is known whether methods exist that could avoid the trade-off between bias and variance. We propose a general strategy to obtain lower bounds on the variance of any estimator with bias smaller than a prespecified bound. This shows to which extent the bias-variance trade-off is unavoidable and allows to quantify the loss of performance for methods that do not obey it. The approach is based on a number of abstract lower bounds for the variance involving the change of expectation with respect to different probability measures as well as information measures such as the Kullback-Leibler or $χ^2$-divergence. In a second part of the article, the abstract lower bounds are applied to several statistical models including the Gaussian white noise model, a boundary estimation problem, the Gaussian sequence model and the high-dimensional linear regression model. For these specific statistical applications, different types of bias-variance trade-offs occur that vary considerably in their strength. For the trade-off between integrated squared bias and integrated variance in the Gaussian white noise model, we propose to combine the general strategy for lower bounds with a reduction technique. This allows us to reduce the original problem to a lower bound on the bias-variance trade-off for estimators with additional symmetry properties in a simpler statistical model. In the Gaussian sequence model, different phase transitions of the bias-variance trade-off occur. Although there is a non-trivial interplay between bias and variance, the rate of the squared bias and the variance do not have to be balanced in order to achieve the minimax estimation rate.

Alexis Derumigny, Jean-David Fermanian

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.