  We propose a likelihood ratio statistic for forming hypothesis tests and
confidence intervals for a nonparametrically estimated univariate regression
function, based on the shape restriction of concavity (alternatively,
convexity). Dealing with the likelihood ratio statistic requires studying an
estimator satisfying a null hypothesis, that is, studying a concave
least-squares estimator satisfying a further equality constraint. We study this
null hypothesis least-squares estimator (NLSE) here, and use it to study our
likelihood ratio statistic. The NLSE is the solution to a convex program, and
we find a set of inequality and equality constraints that characterize the
solution. We also study a corresponding limiting version of the convex program
based on observing a Brownian motion with drift. The solution to the limit
problem is a stochastic process. We study the optimality conditions for the
solution to the limit problem and find that they match those we derived for the
solution to the finite sample problem. This allows us to show the limit
stochastic process yields the limit distribution of the (finite sample) NLSE.
We conjecture that the likelihood ratio statistic is asymptotically pivotal,
meaning that it has a limit distribution with no nuisance parameters to be
estimated, which makes it a very effective tool for this difficult inference
problem. We provide a partial proof of this conjecture, and we also provide
simulation evidence strongly supporting this conjecture.
