  In this work, we delve into the nonparametric empirical Bayes theory and
approximate the classical Bayes estimator by a truncation of the generalized
Laguerre series and then estimate its coefficients by minimizing the prior risk
of the estimator. The minimization process yields a system of linear equations
the size of which is equal to the truncation level. We focus on the empirical
Bayes estimation problem when the mixing distribution, and therefore the prior
distribution, has a support on the positive real half-line or a subinterval of
it. By investigating several common mixing distributions, we develop a strategy
on how to select the parameter of the generalized Laguerre function basis so
that our estimator {possesses a finite} variance. We show that our generalized
Laguerre empirical Bayes approach is asymptotically optimal in the minimax
sense. Finally, our convergence rate is compared and contrasted with {several}
results from the literature.
