We consider M-estimates in a regression model where the noises are of unknown
but stochastically bounded distribution. An asymptotic minimax M-estimate is
derived. Simulations demonstrate the robustness of this approach, as well as
advantages over commonly used estimates such as ordinary least square estimate
and the Huber's estimate. The new method is named regression with tochastically
bounded noises (RSBN). We provide an iterative numerical solution, which is
derived from the proximal point method. The iterative method is elegant,
however may not have fast rate of convergence. RSBN can also be solved by
applying existing state-of-the-art nonlinear optimization software. We present
SNOPT as one example. Insights from RSBN are discussed. (Joint work with Xiaoming Huo).