Local Polynomial Estimation Of A Monotone Regression Function Of Punishment

This paper suggests a penalized local polynomial method to estimate the monotone regression function in nonparametric regression model. In addition to the monotonicity, the newly proposed estimators achieve some desirable balance between monotonicity and indispensable asymptotic properties. Compared to the conventionally used techniques for monotone estimation, the newly proposed method gives some explicit representations for the estimators and then is computationally more efficient. Furthermore, this method can be extended to general case when the regression function and variance function are constrained by their derivatives. For example the newly proposed estimators satisfy the prescribed conditions such as monotonic and/or convex conditions. Some simulations are given to illustrate the theoretical results.In the first chapter of this paper the background and structure are introduced. Chapter 2 outlines the standard local polynomial estimation, consider the nonparametric regression model in the second chapter:Y = r(x) + ε.(4)We locally approximate r(z) in a neighborhood of x by a polynomial asWe then carry out a local polynomial regression by minimizingHere K_ih) = h^-1K((X_i — x)/h), h is a bandwith depending on n, kernel function K(x)is nonnegative and differentiable for x (?) [0,1] and satisfies (2.4) and (2.5). The nonparametric regression estimation can reduce the asymptotic bias and asymptotic variance, adapt automatically to the boundary of design points and have asymptotic normality. However. we can not guarantee that the regression estimator f(x) is monotone when the regression function r(x) is assumed to be monotone. In subsection 3.1, by combining the constraint-based method...