  Shape-constrained inference has wide applicability in bioassay, medicine,
economics, risk assessment, and many other fields. Although there has been a
large amount of work on monotone-constrained univariate curve estimation,
multivariate shape-constrained problems are much more challenging, and fewer
advances have been made in this direction. With a focus on monotone regression
with multiple predictors, this current work proposes a projection approach to
estimate a multiple monotone regression function. An initial unconstrained
estimator -- such as a local polynomial estimator or spline estimator -- is
first obtained, which is then projected onto the shape-constrained space. A
shape-constrained estimate (with multiple predictors) is obtained by
sequentially projecting an (adjusted) initial estimator along each univariate
direction. Compared to the initial unconstrained estimator, the projection
estimate results in a reduction of estimation error in terms of both $L^p$
($p\geq 1$) distance and supremum distance. We also derive the asymptotic
distribution of the projection estimate. Simple computational algorithms are
available for implementing the projection in both the unidimensional and higher
dimensional cases. Our work provides a simple recipe for practitioners to use
in real applications, and is illustrated with a joint-action example from
environmental toxicology.
