  In finite mixtures of location-scale distributions, if there is no constraint
or penalty on the parameters, then the maximum likelihood estimator does not
exist because the likelihood is unbounded. To avoid this problem, we consider a
penalized likelihood, where the penalty is a function of the minimum of the
ratios of the scale parameters and the sample size. It is shown that the
penalized maximum likelihood estimator is strongly consistent. We also analyze
the consistency of a penalized maximum likelihood estimator where the penalty
is imposed on the scale parameters themselves.
