 
Summary: Majorizing estimators and
the approximation of #Pcomplete problems
Leonard J. Schulman \Lambda Vijay V. Vazirani \Lambda
Abstract
A key step in counting via sampling is constructing an un
biased estimator, X, for the parameter ` in question, and
proving a bound on its second moment, E(X 2 ). A key ap
plication of this method is to obtaining a FPRAS for a #P
complete problem; a FPRAS results if the ratio r = E(X 2 ) 1=2
E(X)
is polynomially bounded in the size of the input. We show
that if no additional information is available about the dis
tribution of X, then this condition is also necessary.
The proof involves establishing a new optimality result in
parametric statistics. We introduce the notion of a majoriz
ing estimator, a very strict optimality requirement that we
need for making worstcase (over inputs) and inprobability
(of falling in the desired accuracy range of the parameter `)
statements. We show that for the problem of estimating the
mean of a Gaussian distribution (from the variablelocation,
