** Next:** Degrees of Freedom
** Up:** Deriving the Optimal Scoring
** Previous:** Expected value of the

The variance
can be derived
as follows. With the asymptotic value of *E*[*d*^{*}] we compute the
Taylor series of
(*d*^{*} - *E*[*d*^{*}])^{2} in powers of
(*w*/*n* - *S*_{E}(*d*)).
Then we take expected values, replacing the powers of
(*w*/*n* -
*S*_{E}(*d*))^{k} by the central moments described in equations
8, 9 and 10. Truncating after two
terms we obtain

The most effective estimator is the one among all possible
estimators, which has minimal variance.
The variance is

where
.
We can compute *E* so that it minimizes *F*(*d*).
This will give us asymptotically (in
*n*) optimal estimators. We have several choices for the
minimization: we can derive the best *E* for
- a given distance
*d*, e.g.
.
- a norm for a range of distances,
e.g. for
*d*=0 to 200, i.e.
.
- the minimax for a range of distances, e.g. for
*d*=0 to 200,
i.e
.

There is no hope of finding a closed formula for this optimal
*E*, but the first and second cases can be computed numerically
without much difficulty.

** Next:** Degrees of Freedom
** Up:** Deriving the Optimal Scoring
** Previous:** Expected value of the
*Chantal Korostensky*

*1999-07-14*