  This work presents new cardinality estimation methods for data sets recorded
by HyperLogLog sketches. A simple derivation of the original estimator was
found, that also gives insight how to correct its deficiencies. The result is
an improved estimator that is unbiased over the full cardinality range, is easy
computable, and does not rely on empirically determined data as previous
approaches. Based on the maximum likelihood principle a second unbiased
estimation method is presented which can also be extended to estimate
cardinalities of union, intersection, or relative complements of two sets that
are both represented as HyperLogLog sketches. Experimental results show that
this approach is more precise than the conventional technique using the
inclusion-exclusion principle.
