Optimal sparsity testing in linear regression model

We consider the problem of sparsity testing in high-dimensional linear regression model. The is to test whether number non-zero components (aka sparsity) parameter θ∗ less than or equal k0. pinpoint minimax separation distances for this problem, which amounts quantifying how far a k1-sparse vector has be from set k0-sparse vectors so that able reject null hypothesis with high probability. Two scenarios are considered. In independent scenario, covariates i.i.d. normally distributed and noise level known. general both covariance matrix unknown. Although differ these two scenarios, them actually depend on k0 k1 illustrating composite-composite size alternative hypotheses play key role.