Adaptive minimax test of independence

Yodé, Armel Fabrice

doi:10.3103/s1066530711030069

Cited by 3 publications

(5 citation statements)

References 7 publications

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“…The proof of this theorem is partly inspired by the lower bound established in Yodé [22] and Yodé [24]. Let us fix σ > 0 and put δ n = σh n and M n = δ −1 n .…”

Section: Proof Of Lower Boundmentioning

confidence: 97%

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Asymptotically Minimax Goodness-of-fit Testing for Single-index Models

Tchiekre¹,

Pouet²,

Yode³

2022

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In the context of non parametric multivariate regression model, we are interested in goodness-of-fit testing for the single-index models. These models are dimension reduction models and are therefore useful in multidimensional nonparametric statistics because of the well-known phenomenon called the curse of dimensionality. Fan and Li [5] have proposed the first consistent test for goodness-of-fit testing of the single-index by using nonparametric kernel estimation method and a central limit theorem for degenerate U -statistics of order higher than two. Since then, the minimax properties of this test have not been investigated. Following this work, we use here the asymptotic minimax approach. We are interested in finding the asymptotic minimax rate of testing φ(α n ) which gives the minimal distance between the null and alternative hypotheses such that a successful testing is possible. We propose a test procedure of level α n which can tend to zero when the sample size tends to infinity. We have established the minimax asymptotic properties of our test procedure by showing that it reaches the asymptotic minimax rate φ n (α n ) for the dimension d = 3 and there is no test of level α n reaching this rate for d ≥ 3. Because of its minimax asymptotic properties, our test is able to distinguish the null hypothesis of the closest possible alternative. The results obtained were possible thanks to a large deviation result that we established for a degenerate U -statistic of order two appearing in our decision variable.

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“…The proof of this theorem is partly inspired by the lower bound established in Yodé [22] and Yodé [24]. Let us fix σ > 0 and put δ n = σh n and M n = δ −1 n .…”

Section: Proof Of Lower Boundmentioning

confidence: 97%

“…We show also how this property affects the error of the second kind. This condition has already been used in Yodé [22,24] and Chiabrando [2]. Such kind of testing problems appear in the concept of random normalizing factor initiated by Lepski [16] and extended by Hoffmann and Lepski [11], Hoffman [10], Yodé [23], and Chiabrando [2].…”

Section: Introductionmentioning

confidence: 99%

Asymptotically Minimax Goodness-of-fit Testing for Single-index Models

Tchiekre¹,

Pouet²,

Yode³

2022

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“…However, only few works exist for the problem of minimax independence testing. The notable works are those of Ingster [Ingster, 1989, Ingster, 1993b and of Yodé [Yodé, 2004, Yodé, 2011 in the asymptotic framework, and the one of Albert [Albert, 2015] in the non-asymptotic framework considered in this paper. As far as we know, no lower-bound for the minimax rate of testing independence was yet proved in the non-asymptotic framework.…”

Section: Adaptive Hsic-based Independence Testsmentioning

confidence: 99%

“…We mention for instance [Baraud et al, 2003] for linear regression model testing with normal noise and [Butucea and Tribouley, 2006] for testing the equality of two samples densities. For the specific case of testing independence, the adaptive testing procedure introduced in [Yodé, 2011] seems to be the only currently existing. As mentioned above, this test is purely asymptotic, but we are interested here in the non-asymptotic framework.…”

Section: Adaptive Hsic-based Independence Testsmentioning

confidence: 99%

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Adaptive test of independence based on HSIC measures

et al. 2022

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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 α ∈ (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.

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