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Motivated by a neuroscience question about synchrony detection in spike train analysis, we deal with the independence testing problem for point processes. We introduce nonparametric test statistics, which are rescaled general U-statistics, whose corresponding critical values are constructed from bootstrap and randomization/permutation approaches, making as few assumptions as possible on the underlying distribution of the point processes. We derive general consistency results for the bootstrap and for the permutation w.r.t. Wasserstein's metric, which induces weak convergence as well as convergence of second-order moments. The obtained bootstrap or permutation independence tests are thus proved to be asymptotically of the prescribed size, and to be consistent against any reasonable alternative. A simulation study is performed to illustrate the derived theoretical results, and to compare the performance of our new tests with existing ones in the neuroscientific literature.

Initially motivated by the study of the non-asymptotic properties of non-parametric tests based on permutation methods, concentration inequalities for uniformly permuted sums have been largely studied in the literature. Recently, Delyon et al. proved a new Bernstein-type concentration inequality based on martingale theory. This work presents a new proof of this inequality based on the fundamental inequalities for random permutations of Talagrand. The idea is to first obtain a rough inequality for the square root of the permuted sum, and then, iterate the previous analysis and plug this first inequality to obtain a general concentration of permuted sums around their median. Then, concentration inequalities around the mean are deduced. This method allows us to obtain the Bernstein-type inequality up to constants, and, in particular, to recovers the Gaussian behavior of such permuted sums under classical conditions encountered in the literature. Then, an application to the study of the second kind error rate of permutation tests of independence is presented.

We consider the problem of nonparametric density estimation under privacy constraints in an adversarial framework. To this end, we study minimax rates under local differential privacy over Sobolev spaces. We first obtain a lower bound which allows us to quantify the impact of privacy compared with the classical framework. Next, we introduce a new Coordinate block privacy mechanism that guarantees local differential privacy, which, coupled with a projection estimator, achieves the minimax optimal rates.

We propose two novel nonparametric two-sample kernel tests based on the Maximum Mean Discrepancy (MMD). First, for a fixed kernel, we construct an MMD test using either permutations or a wild bootstrap, two popular numerical procedures to determine the test threshold. We prove that this test controls the probability of type I error non-asymptotically. Hence, it can be used reliably even in settings with small sample sizes as it remains well-calibrated, which differs from previous MMD tests which only guarantee correct test level asymptotically. When the difference in densities lies in a Sobolev ball, we prove minimax optimality of our MMD test with a specific kernel depending on the smoothness parameter of the Sobolev ball. In practice, this parameter is unknown and, hence, the optimal MMD test with this particular kernel cannot be used. To overcome this issue, we construct an aggregated test, called MMDAgg, which is adaptive to the smoothness parameter. The test power is maximised over the collection of kernels used, without requiring held-out data for kernel selection (which results in a loss of test power), or arbitrary kernel choices such as the median heuristic. We prove that MMDAgg still controls the level non-asymptotically, and achieves the minimax rate over Sobolev balls, up to an iterated logarithmic term. Our guarantees are not restricted to a specific type of kernel, but hold for any product of one-dimensional translation invariant characteristic kernels. We provide a user-friendly parameter-free implementation of MMDAgg using an adaptive collection of bandwidths. We demonstrate that MMDAgg significantly outperforms alternative state-of-the-art MMD-based two-sample tests on synthetic data satisfying the Sobolev smoothness assumption, and that, on real-world image data, MMDAgg closely matches the power of tests leveraging the use of models such as neural networks.

Dependence measures based on reproducing kernel Hilbert spaces, also known as Hilbert-Schmidt Independence Criterion and denoted HSIC, are widely used to statistically decide whether or not two random vectors are dependent. Recently, non-parametric HSIC-based statistical tests of independence have been performed. However, these tests lead to the question of the choice of the kernels associated to the HSIC. In particular, there is as yet no method to objectively select specific kernels with theoretical guarantees in terms of first and second kind errors. One of the main contributions of this work is to develop a new HSIC-based aggregated procedure which avoids such a kernel choice, and to provide theoretical guarantees for this procedure. To achieve this, we first introduce non-asymptotic single tests based on Gaussian kernels with a given bandwidth, which are of prescribed level \\(\\alpha \\in (0,1)\\). From a theoretical point of view, we upper-bound their uniform separation rate of testing over Sobolev and Nikol'skii balls. Then, we aggregate several single tests, and obtain similar upper-bounds for the uniform separation rate of the aggregated procedure over the same regularity spaces. Another main contribution is that we provide a lower-bound for the non-asymptotic minimax separation rate of testing over Sobolev balls, and deduce that the aggregated procedure is adaptive in the minimax sense over such regularity spaces. Finally, from a practical point of view, we perform numerical studies in order to assess the efficiency of our aggregated procedure and compare it to existing independence tests in the literature.

Motivated by a neuroscience question about synchrony detection in spike train analysis, we deal with the independence testing problem for point processes. We introduce non-parametric test statistics, which are rescaled general \\(U\\)-statistics, whose corresponding critical values are constructed from bootstrap and randomization/permutation approaches, making as few assumptions as possible on the underlying distribution of the point processes. We derive general consistency results for the bootstrap and for the permutation w.r.t. to Wasserstein's metric, which induce weak convergence as well as convergence of second order moments. The obtained bootstrap or permutation independence tests are thus proved to be asymptotically of the prescribed size, and to be consistent against any reasonable alternative. A simulation study is performed to illustrate the derived theoretical results, and to compare the performance of our new tests with existing ones in the neuroscientific literature.

We investigate several distribution free dependence detection procedures, mainly based on bootstrap principles and their approximation properties. Thanks to this study, we introduce a new distribution free Unitary Events (UE) method, named Permutation UE, which consists in a multiple testing procedure based on permutation and delayed coincidence count. Each involved single test of this procedure achieves the prescribed level, so that the corresponding multiple testing procedure controls the False Discovery Rate (FDR), and this with as few assumptions as possible on the underneath distribution. Some simulations show that this method outperforms the trial-shuffling and the MTGAUE method in terms of single levels and FDR, for a comparable amount of false negatives. Application on real data is also provided.