A note on estimating a partly linear model under monotonicity constraints

We consider asymptotic properties of the least-squares estimator of a partly linear regression model when the nonparametric component is subject to monotonicity constraints. We show that the least-squares estimator of the finite-dimensional regression coefficient is root-n consistent and asymptotically normal. We also show that the isotonic estimator of the monotone nonparametric function at a fixed point is cube root-n consistent, and apart from a scale constant, has the same limiting distribution in nonparametric monotone density estimation and isotonic regression derived by Prakasa Rao (Sankhya Ser. A 31 (1969) 23) and Brunk (In: M.L. Puri (Ed.), Nonparametric Techniques in Statistical Inference, Cambridge University Press, Cambridge, 1970).