  We consider the problem of choosing a density estimate from a set of
distributions F, minimizing the L1-distance to an unknown distribution
(Devroye, Lugosi 2001). Devroye and Lugosi analyze two algorithms for the
problem: Scheffe tournament winner and minimum distance estimate. The Scheffe
tournament estimate requires fewer computations than the minimum distance
estimate, but has strictly weaker guarantees than the latter.
  We focus on the computational aspect of density estimation. We present two
algorithms, both with the same guarantee as the minimum distance estimate. The
first one, a modification of the minimum distance estimate, uses the same
number (quadratic in |F|) of computations as the Scheffe tournament. The second
one, called ``efficient minimum loss-weight estimate,'' uses only a linear
number of computations, assuming that F is preprocessed.
  We also give examples showing that the guarantees of the algorithms cannot be
improved and explore randomized algorithms for density estimation.
