In this paper, we study the $k$-th moment of central values family quadratic Dirichlet $L$-functions moduli $8p$, with $p$ ranging over odd primes. Assuming truth generlized Riemann hypothesis, establish sharp upper and lower bounds for power these $L$-values all real $k \geq 0$.

We obtain estimation error rates for estimators obtained by aggregation of regularized median-of-means tests, following a construction of Le Cam. The results hold with exponentially large probability, under only weak moments assumptions on data. Any norm may be used for regularization. When it has some sparsity inducing power we recover sparse rates of convergence. The procedure is robust since...