  We consider the least-squares regression problem with unknown noise variance,
where the observed data points are allowed to be corrupted by outliers.
Building on the median-of-means (MOM) method introduced by Lecue and Lerasle
Ann.Statist.48(2):906-931(April 2020) in the case of known noise variance, we
propose a general MOM approach for simultaneous inference of both the
regression function and the noise variance, requiring only an upper bound on
the noise level. Interestingly, this generalization requires care due to
regularity issues that are intrinsic to the underlying convex-concave
optimization problem. In the general case where the regression function belongs
to a convex class, we show that our simultaneous estimator achieves with high
probability the same convergence rates and a similar risk bound as if the noise
level was unknown, as well as convergence rates for the estimated noise
standard deviation.
  In the high-dimensional sparse linear setting, our estimator yields a robust
analog of the square-root LASSO. Under weak moment conditions, it jointly
achieves with high probability the minimax rates of estimation $s^{1/p}
\sqrt{(1/n) \log(p/s)}$ for the $\ell_p$-norm of the coefficient vector, and
the rate $\sqrt{(s/n) \log(p/s)}$ for the estimation of the noise standard
deviation. Here $n$ denotes the sample size, $p$ the dimension and $s$ the
sparsity level. We finally propose an extension to the case of unknown sparsity
level $s$, providing a jointly adaptive estimator $(\widetilde \beta,
\widetilde \sigma, \widetilde s)$. It simultaneously estimates the coefficient
vector, the noise level and the sparsity level, with proven bounds on each of
these three components that hold with high probability.
