  We consider the problem of constructing pointwise confidence intervals in the
multiple isotonic regression model. Recently, [HZ19] obtained a pointwise limit
distribution theory for the so-called block max-min and min-max estimators
[FLN17] in this model, but inference remains a difficult problem due to the
nuisance parameter in the limit distribution that involves multiple unknown
partial derivatives of the true regression function.
  In this paper, we show that this difficult nuisance parameter can be
effectively eliminated by taking advantage of information beyond point
estimates in the block max-min and min-max estimators. Formally, let
$\hat{u}(x_0)$ (resp. $\hat{v}(x_0)$) be the maximizing lower-left (resp.
minimizing upper-right) vertex in the block max-min (resp. min-max) estimator,
and $\hat{f}_n$ be the average of the block max-min and min-max estimators. If
all (first-order) partial derivatives of $f_0$ are non-vanishing at $x_0$, then
the following pivotal limit distribution theory holds: $$
\sqrt{n_{\hat{u},\hat{v}}(x_0)}\big(\hat{f}_n(x_0)-f_0(x_0)\big)\rightsquigarrow
\sigma\cdot \mathbb{L}_{1_d}. $$ Here $n_{\hat{u},\hat{v}}(x_0)$ is the number
of design points in the block $[\hat{u}(x_0),\hat{v}(x_0)]$, $\sigma$ is the
standard deviation of the errors, and $\mathbb{L}_{1_d}$ is a universal limit
distribution free of nuisance parameters. This immediately yields confidence
intervals for $f_0(x_0)$ with asymptotically exact confidence level and oracle
length. Notably, the construction of the confidence intervals, even new in the
univariate setting, requires no more efforts than performing an isotonic
regression for once using the block max-min and min-max estimators, and can be
easily adapted to other common monotone models. Extensive simulations are
carried out to support our theory.
