  We derive quantitative bounds on the rate of convergence in $L^1$ Wasserstein
distance of general M-estimators, with an almost sharp (up to a logarithmic
term) behavior in the number of observations. We focus on situations where the
estimator does not have an explicit expression as a function of the data. The
general method may be applied even in situations where the observations are not
independent. Our main application is a rate of convergence for cross validation
estimation of covariance parameters of Gaussian processes.
