Elaborating on the work of Ibragimov and Has'minskii (1981) we prove a law of large deviations (LLD) for $M$-estimators, i.e., those estimators which maximize a functional, continuous in the parameter, of the observations. This LLD is applied, using the results of Petrov (1975), to the problem of parametrical nonlinear regression in the situation of discrete time, independent errors and regression functions which are continuous in the parameter. This improves a result of Prakasa Rao (1984).
"A Large Deviation Result for Parameter Estimators and its Application to Nonlinear Regression Analysis." Ann. Statist. 15 (3) 1031 - 1049, September, 1987. https://doi.org/10.1214/aos/1176350491