On consistency of the MLE.
František Rublík · Czech digital mathematics library · 1995
Convergence of the maximum likelihood estimator is established without the assumption that the true value of the parameter belongs to the null hypothesis QQ.It is shown, that the MLE exists with probability tending to 1, and that the distance of the MLE from a set H of parameters from Qo tends to zero almost everywhere, where H are parameters of the probabilities best fitting the true distribution in the sense that they maximize the mean of logarithm of the likelihood function.