Confidence bands for convex median curves using sign-tests

Abstract

Suppose that one observes pairs $(x_1,Y_1)$, $(x_2,Y_2)$, \ldots, $(x_n,Y_n)$, where $x_1 \le x_2 \le \cdots \le x_n$ are fixed numbers, and $Y_1, Y_2, \ldots, Y_n$ are independent random variables with unknown distributions. The only assumption is that ${\rm Median}(Y_i) = f(x_i)$ for some unknown convex function $f$. We present a confidence band for this regression function $f$ using suitable multiscale sign-tests. While the exact computation of this band requires $O(n^4)$ steps, good approximations can be obtained in $O(n^2)$ steps. In addition the confidence band is shown to have desirable asymptotic properties as the sample size $n$ tends to infinity.