An RD-space $\mathcal X$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type $\mathcal X$ having “dimension” $n$, there exists a $p_0\in (n/(n+1), 1)$ such that for certain es of distributions, the $L^p(\mathcal X)$ quasi-norms of their radial maximal functions and grand maximal functions are equivalent when $p\in (p_0,\infty ]$. This result yields a radial maximal function characterization for Hardy spaces on $\mathcal X$.