  A common problem in econometrics, statistics, and machine learning is to
estimate and make inference on functions that satisfy shape restrictions. For
example, distribution functions are nondecreasing and range between zero and
one, height growth charts are nondecreasing in age, and production functions
are nondecreasing and quasi-concave in input quantities. We propose a method to
enforce these restrictions ex post on point and interval estimates of the
target function by applying functional operators. If an operator satisfies
certain properties that we make precise, the shape-enforced point estimates are
closer to the target function than the original point estimates and the
shape-enforced interval estimates have greater coverage and shorter length than
the original interval estimates. We show that these properties hold for six
different operators that cover commonly used shape restrictions in practice:
range, convexity, monotonicity, monotone convexity, quasi-convexity, and
monotone quasi-convexity. We illustrate the results with two empirical
applications to the estimation of a height growth chart for infants in India
and a production function for chemical firms in China.
