  For the Gaussian sequence model, we obtain non-asymptotic minimax rates of
estimation of the linear, quadratic and the L2-norm functionals on classes of
sparse vectors and construct optimal estimators that attain these rates. The
main object of interest is the class s-sparse vectors for which we also provide
completely adaptive estimators (independent of s and of the noise variance)
having only logarithmically slower rates than the minimax ones. Furthermore, we
obtain the minimax rates on the Lq-balls where 0 < q < 2. This analysis shows
that there are, in general, three zones in the rates of convergence that we
call the sparse zone, the dense zone and the degenerate zone, while a fourth
zone appears for estimation of the quadratic functional. We show that, as
opposed to estimation of the vector, the correct logarithmic terms in the
optimal rates for the sparse zone scale as log(d/s^2) and not as log(d/s). For
the sparse class, the rates of estimation of the linear functional and of the
L2-norm have a simple elbow at s = sqrt(d) (boundary between the sparse and the
dense zones) and exhibit similar performances, whereas the estimation of the
quadratic functional reveals more complex effects and is not possible only on
the basis of sparsity described by the sparsity condition on the vector.
Finally, we apply our results on estimation of the L2-norm to the problem of
testing against sparse alternatives. In particular, we obtain a non-asymptotic
analog of the Ingster-Donoho-Jin theory revealing some effects that were not
captured by the previous asymptotic analysis.
