A note on estimating a smooth monotone regression by combining kernel and density estimates

Abstract

In a recent paper Dette, Neumeyer and Pilz (H. Dette, N. Neumeyer, and K.F. Pilz, A simple nonparametric estimator of a strictly monotone regression function, Bernoulli 12 (2006), pp. 469-490) proposed a new nonparametric estimate of a smooth monotone regression function. This method is based on a nondecreasing rearrangement of an arbitrary unconstrained nonparametric estimator. Under the assumption of a twice continuously differentiable regression function, the estimate is first-order asymptotic equivalent to the unconstrained estimate and other types of smooth monotone estimates. In this note, we provide a more refined asymptotic analysis of the monotone regression estimate. In the case where the regression function is increasing but only once continuously differentiable, we prove the asymptotic normality of an appropriately standardised version of the estimate, where the asymptotic variance is of order n −2/3−ε , the bias is of order n −1/3+ε and ε > 0 is small. Therefore, the rate of convergence of the new estimate is worse than the rate n −1/3 of the (unsmoothed) monotone least squares estimate. On the other hand, if the derivative of the regression function is Hölder continuous, the rate of convergence of the new estimate is faster than n −1/3 .

Shape constrained kernel density estimation

“…In a first step, the density is estimated by an unconstrained kernel estimate and then, in a second step, it is modified with respect to the shape constraints by using monotone, convex or concave rearrangements which have been introduced in nonparametric regression by Dette, Neumeyer and Pilz (2006), Birke and Dette (2005) and Birke and Dette (2007). Those methods produce shape constrained estimators which have the same asymptotic distribution as the unconstrained estimator one starts with.…”

The independence process in conditional quantile location-scale models and an application to testing for monotonicity

“…Remark 4.2 Note that we use non-smooth monotone rearrangement estimatorsq τ,I . Dette et al (2006) and Birke and Dette (2008) consider smooth versions of the increasing rearrangements in the context of monotone mean regression. Corresponding increasing quantile curve estimators could be defined aŝ…”

Testing Monotonicity of Regression Functions – An Empirical Process Approach

“…In the following we specifically consider the estimator by Dette et al. (2006) and give simple conditions under which assumptions (A6)–(A8) are fulfilled (see also Birke & Dette, 2008 for further discussion of this estimator). The estimator is defined as the generalized inverse of , that is where…”