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## The doubly censored'' case

For the entropy estimation (in section 5.3), it is necessary to estimate and, consequently, to use information from distances . The likelihood function for the doubly censored set of data is given by (cf. [Kendall and Stuart, 1979, eq. 32.37]):
 (5.22)

The likelihood equations are:
 (5.23)

and
 (5.24)

so that
 (5.25)

Note that depends on the reference distance. Substitution of eq. (5.23) into (5.24) and dividing all distances by yields:
 (5.26)

which is very similar to Ellner's result. The difference is that the total number of distances () is fixed here instead of . (That is why we had to use eq. (5.23) as well). The asymptotic variance of maximum likelihood estimators of a function of parameters (here and ) is given by [Kendall and Stuart, 1979, eq. 17.87]:
 (5.27)

where is the information matrix
 (5.28)

Like eq. (5.9), eq. (5.27) takes account only of terms in the variance [Kendall and Stuart, 1979, §17.24]. Solving eq. (5.27) for yields:
 (5.29)

The elements of the information matrix are obtained from the second partial derivatives of the likelihood function:
 (5.30)

 (5.31)

 (5.32)

Thus
 (5.33)

 (5.34)

 (5.35)

since , and . Note that
 (5.36)

Now
 (5.37)

If we substitute the estimated values and we obtain:
 (5.38)

which is once again similar to Ellner's result. Thus, asymptotically (because is fixed instead of ), the doubly censored'' estimator of the dimension is the same as Ellner's.

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