  Let $(Y_i,\theta_i)$, $i=1,...,n$, be independent random vectors distributed
like $(Y,\theta) \sim G^*$, where the marginal distribution of $\theta$ is
completely unknown, and the conditional distribution of $Y$ conditional on
$\theta$ is known. It is desired to estimate the marginal distribution of
$\theta$ under $G^*$, as well as functionals of the form $E_{G^*} h(Y,\theta)$
for a given $h$, based on the observed $Y_1,...,Y_n$.
  In this paper we suggest a deconvolution method for the above estimation
problems and discuss some of its applications in Empirical Bayes analysis. The
method involves a quadratic programming step, which is an elaboration on the
formulation and technique in Efron(2013). It is computationally efficient and
may handle large data sets, where the popular method, of deconvolution using
EM-algorithm, is impractical.
  The main application that we study is treatment of non-response. Our approach
is nonstandard and does not involve missing at random type of assumptions. The
method is demonstrated in simulations, as well as in an analysis of a real data
set from the Labor force survey in Israel. Other applications including
estimation of the risk, and estimation of False Discovery Rates, are also
discussed.
  We also present a method, that involves convex optimization, for constructing
confidence intervals for $E_{G^*} h$, under the above setup.
