  We present a unified approach to goodness-of-fit testing in $\mathbb{R}^d$
and on lower-dimensional manifolds embedded in $\mathbb{R}^d$ based on sums of
powers of weighted volumes of $k$-th nearest neighbor spheres. We prove
asymptotic normality of a class of test statistics under the null hypothesis
and under fixed alternatives. Under such alternatives, scaled versions of the
test statistics converge to the $\alpha$-entropy between probability
distributions. A simulation study shows that the procedures are serious
competitors to established goodness-of-fit tests.
