Discontinuous versus smooth regression

Abstract

Z.Given measurements $(x_i,y_i) = 1,\dots,n$, we discuss methods to assess whether an underlying regression function is smooth (continuous or differentiable) or whether it has discontinuities. The variance of the measurements is assumed to be unknown, and is estimated simultane-ously. By regressing squared differences of the data formed with various span sizes on the span size itself, we obtain an asymptotic linear model with dependent errors. The parameters of this asymptotic linear model include the sum of the squared jump sizes as well as the variance of the measurements. Both parameters can be consistently estimated, with mean squared error rates of convergence of $n^{-2/3}$ for the sum of squared jump sizes and $n^{-1}$ for the error variance. We derive the asymptotic constants of the mean squared error (MSE) and discuss the dependence of MSE on the maximum span size $L$. The test for the existence of jumps is formulated for the null hypothesis that the sum of squared jump sizes is 0. The asymptotic distribution of the test statistic is obtained essentially via a central limit theorem for $U$-statistics. We motivate and illustrate the methods with data surrounded by a scientific controversy concerning the question whether the growth of children occurs smoothly or rather in jumps.