Distribution-free performance bounds for potential function rules

In the discrimination problem the random variable
\theta
, known to take values in
{1, \cdots ,M}
, is estimated from the random vector
X
. All that is known about the joint distribution of
(X, \theta)
is that which can be inferred from a sample
(X_{1}, \theta_{1}), \cdots ,(X_{n}, \theta_{n})
of size
n
drawn from that distribution. A discrimination nde is any procedure which determines a decision
\hat{ \theta}
for
\theta
from
X
and
(X_{1}, \theta_{1}) , \cdots , (X_{n}, \theta_{n})
. For rules which are determined by potential functions it is shown that the mean-square difference between the probability of error for the nde and its deleted estimate is bounded by
A/ \sqrt{n}
where
A
is an explicitly given constant depending only on
M
and the potential function. The
O(n ^{-1/2})
behavior is shown to be the best possible for one of the most commonly encountered rules of this type.