  Let $p$ be a prime $\ge 5$. We establish explicit rates of overconvergence
for members of the "Eisenstein family", notably for the $p$-adic modular
function $V(E_{(1,0)}^{\ast})/E_{(1,0)}^{\ast}$ ($V$ the $p$-adic Frobenius
operator) that plays a pi\-votal role in Coleman's theory of $p$-adic families
of modular forms. The proof goes via an in-depth analysis of rates of
overconvergence of $p$-adic modular functions of form $V(E_k)/E_k$ where $E_k$
is the classical Eisenstein series of level $1$ and weight $k$ divisible by
$p-1$. Under certain conditions, we extend the latter result to a vast
generalization of a theorem of Coleman--Wan regarding the rate of
overconvergence of $V(E_{p-1})/E_{p-1}$. We also comment on previous results in
the literature. These include applications of our results for the primes $5$
and $7$.
