Suppose that $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_n$ are two independent samples following a Gaussian law on $\mathbb{R}^p$ with zero mean and unknown invertible covariances $A$ and $B$. We consider the situation where $p$ is much larger than $n$ but we assume that $A^{-1}$ and $B^{-1}$ are sparse (conditional independence on the components). How can we test efficiently if $A=B$? Same question if the sparsity structure is assumed in $A$ and $B$ rather than on their inverse (independence instead of conditional independence).
Suppose that $(X_1,Y_1),\ldots,(X_n,Y_n)$ is a sample drawn from some unknown law on $\mathbb{R}^2.$ How can we efficiently test if the marginal laws are identical? More generally, suppose that $\ldots,Z_{-1}, Z_0, Z_1,\ldots$ is a time series. How can we test if the series is stationary? In other words, how can we test a nonparametric linear structure from the observation of a sample of an unknown law? Random projections?
For these questions coming from applications, we seek for a concrete usable answer…
This post is inspired from (separate) informal discussions with Georges Oppenheim and Didier Concordet. My friend and colleague Christophe Giraud told me that he is working with Nicolas Verzelen and Fanny Villers on the first question.
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