  Let $X_1,\ldots,X_n$ be a random sample from the Gamma distribution with
density $f(x)=\lambda^{\alpha}x^{\alpha-1}e^{-\lambda x}/\Gamma(\alpha)$,
$x>0$, where both $\alpha>0$ (the shape parameter) and $\lambda>0$ (the
reciprocal scale parameter) are unknown. The main result shows that the
uniformly minimum variance unbiased estimator (UMVUE) of the shape parameter,
$\alpha$, exists if and only if $n\geq 4$; moreover, it has finite variance if
and only if $n\geq 6$. More precisely, the form of the UMVUE is given for all
parametric functions $\alpha$, $\lambda$, $1/\alpha$ and $1/\lambda$.
Furthermore, a highly efficient estimating procedure for the two-parameter Beta
distribution is also given. This is based on a Stein-type covariance identity
for the Beta distribution, followed by an application of the theory of
$U$-statistics and the delta-method.
  MSC: Primary 62F10; 62F12; Secondary 62E15.
  Key words and phrases: unbiased estimation; Gamma distribution; Beta
distribution; Ye-Chen-type closed-form estimators; asymptotic efficiency;
$U$-statistics; Stein-type covariance identity; delta-method.
