  We study limit distributions for the tuning-free max-min block estimator
originally proposed in [FLN17] in the problem of multiple isotonic regression,
under both fixed lattice design and random design settings. We show that, if
the regression function $f_0$ admits vanishing derivatives up to order
$\alpha_k$ along the $k$-th dimension ($k=1,\ldots,d$) at a fixed point $x_0
\in (0,1)^d$, and the errors have variance $\sigma^2$, then the max-min block
estimator $\hat{f}_n$ satisfies \begin{align*}
(n_\ast/\sigma^2)^{\frac{1}{2+\sum_{k \in \mathcal{D}_\ast}
\alpha_k^{-1}}}\big(\hat{f}_n(x_0)-f_0(x_0)\big)\rightsquigarrow
\mathbb{C}(f_0,x_0). \end{align*} Here $\mathcal{D}_\ast, n_\ast$, depending on
$\{\alpha_k\}$ and the design points, are the set of all `effective dimensions'
and the size of `effective samples' that drive the asymptotic limiting
distribution, respectively. If furthermore either $\{\alpha_k\}$ are relative
primes to each other or all mixed derivatives of $f_0$ of certain critical
order vanish at $x_0$, then the limiting distribution can be represented as
$\mathbb{C}(f_0,x_0) =_d K(f_0,x_0) \cdot \mathbb{D}_{\alpha}$, where
$K(f_0,x_0)$ is a constant depending on the local structure of the regression
function $f_0$ at $x_0$, and $\mathbb{D}_{\alpha}$ is a non-standard limiting
distribution generalizing the well-known Chernoff distribution in univariate
problems. The above limit theorem is also shown to be optimal both in terms of
the local rate of convergence and the dependence on the unknown regression
function whenever such dependence is explicit (i.e. $K(f_0,x_0)$), for the full
range of $\{\alpha_k\}$ in a local asymptotic minimax sense.
