  For high dimensional statistical models, researchers have begun to focus on
situations which can be described as having relatively few moderately large
coefficients. Such situations lead to some very subtle statistical problems. In
particular, Ingster and Donoho and Jin have considered a sparse normal means
testing problem, in which they described the precise demarcation or detection
boundary. Meinshausen and Rice have shown that it is even possible to estimate
consistently the fraction of nonzero coordinates on a subset of the detectable
region, but leave unanswered the question of exactly in which parts of the
detectable region consistent estimation is possible. In the present paper we
develop a new approach for estimating the fraction of nonzero means for
problems where the nonzero means are moderately large. We show that the
detection region described by Ingster and Donoho and Jin turns out to be the
region where it is possible to consistently estimate the expected fraction of
nonzero coordinates. This theory is developed further and minimax rates of
convergence are derived. A procedure is constructed which attains the optimal
rate of convergence in this setting. Furthermore, the procedure also provides
an honest lower bound for confidence intervals while minimizing the expected
length of such an interval. Simulations are used to enable comparison with the
work of Meinshausen and Rice, where a procedure is given but where rates of
convergence have not been discussed. Extensions to more general Gaussian
mixture models are also given.
