Estimation of parameters of probabilistic models which is based on the number of different elements in a sample

We consider a sample from a one-parameter family of distributions on natural numbers. It is assumed that the values of the elements in the sample are unknown and only the number of different elements of the sample is known. From these statistics, we construct estimators of the parameter for a few concrete parametric families. It is shown by an example that such an estimator need not be asymptotically normal.