  We introduce a new methodology for analyzing serial data by quantile
regression assuming that the underlying quantile function consists of constant
segments. The procedure does not rely on any distributional assumption besides
serial independence. It is based on a multiscale statistic, which allows to
control the (finite sample) probability for selecting the correct number of
segments S at a given error level, which serves as a tuning parameter. For a
proper choice of this parameter, this tends exponentially fast to the true S,
as sample size increases. We further show that the location and size of
segments are estimated at minimax optimal rate (compared to a Gaussian setting)
up to a log-factor. Thereby, our approach leads to (asymptotically) uniform
confidence bands for the entire quantile regression function in a fully
nonparametric setup. The procedure is efficiently implemented using dynamic
programming techniques with double heap structures, and software is provided.
Simulations and data examples from genetic sequencing and ion channel
recordings confirm the robustness of the proposed procedure, which at the same
hand reliably detects changes in quantiles from arbitrary distributions with
precise statistical guarantees.
