Confidence bands for convex median curves using sign-tests

Details Confidence bands for convex median curves using sign-tests Confidence bands for convex median curves using sign-tests

2006-09-11

arxiv.org/abs/math/0609285v2

IMS Lecture Notes Monograph Series 2007, Vol. 55, 85-100

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/074921707000000283 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/p

Scientific paper

10.1214/074921707000000283

Suppose that one observes pairs $(x_1,Y_1)$, $(x_2,Y_2)$, ..., $(x_n,Y_n)$, where $x_1\le x_2\le ... \le x_n$ are fixed numbers, and $Y_1,Y_2,...,Y_n$ 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.

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