  Let $\mathcal{B}$ denote a nonempty translation invariant collection of
intervals in $\mathbb{R}^n$ (which we regard as a rare basis), and define the
associated geometric maximal operator $M_\mathcal{B}$ by $$M_\mathcal{B}f(x) =
\sup_{x \in R \in \mathcal{B}} \frac{1}{|R|}\int_R |f|.$$ We provide a
sufficient condition on $\mathcal{B}$ so that the estimate $$ |\{x \in
\mathbb{R}^n : M_{\mathcal{B}}f(x) > \alpha\}|\leq C_n \int_{\mathbb{R}^{n}}
\frac{|f|}{\alpha}\left(1+\log^+\frac{|f|}{\alpha}\right)^{n-1} $$ is sharp. As
a corollary we obtain sharp weak type estimates for maximal operators
associated to several classes of rare bases including C\'ordoba, Soria and
Zygmund bases.
