Now suppose we have a sample that consists of
independent distances, with in the
interval , in and
in .
The likelihood function for this doubly censored
set of data is [Kendall and Stuart, 1979]

By solving the likelihood equations [Kendall and Stuart, 1979], we find
the maximum likelihood estimators of the
parameters , and
. These are

(9) |

where we used equation (3) and the property that a function of maximum likelihood estimators is itself a maximum likelihood estimator.

The asymptotic variances of the dimension and entropy estimators
are obtained by inverting the information matrix [Kendall and Stuart, 1979]. We find

If , then and are uncorrelated. Moreover, the estimator of the entropy is equivalent to equation (6) if one substitutes and if the correlation integrals are based on independent distances.

The maximum likelihood estimator of the correlation dimension
calculated at a single embedding dimension reads

These equations are slight generalizations of Ellner's results. Note that the expressions for the ``double'' correlation dimension (equation (8)) and entropy (equation (10)) are only meaningful if the ``single'' correlation dimensions and (equation (13)) do not significantly differ.