Does anyone know if there are any standard non-asymptotic results for M-estimators? I'm looking for finite-sample guarantees. Figured maybe someone here might know.



Non-asymptotic results for M-estimators?
Uniform Berry--Esseen-type bounds on the rate of convergence to normality for $M$-estimators were obtained by Michel and Phanzagl -- see e.g. Refs. [14, 16, 17] in [this paper](https://projecteuclid.org/download/pdfview_1/euclid.ejs/1491897618). In the latter paper, a non-uniform Berry--Esseen-type bound for maximum likelihood estimators was given as well, with sufficiently detailed instructions (on page 1176) on how to the extend this result to $M$-estimators. 


