Abstract:

Convergence of the maximum likelihood estimator is established without the assumption that the true value of the parameter belongs to the null hypothesis $\Omega _0$. 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 $\Omega _0$ 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.