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).<\/p>\n
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?<\/p>\n
For these questions coming from applications, we seek for a concrete usable answer...<\/p>\n
This post is inspired from (separate) informal discussions with Georges Oppenheim<\/a> and Didier Concordet<\/a>. My friend and colleague Christophe Giraud<\/a> told me that he is working with Nicolas Verzelen<\/a> and Fanny Villers<\/a> on the first question.<\/p>\n","protected":false},"excerpt":{"rendered":" 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 $B$.…<\/p>\n