# Multiscale quantile segmentation

## TL;DR

This paper presents a multiscale quantile segmentation method for serial data that accurately detects change points in quantile functions with strong statistical guarantees, without relying on distributional assumptions beyond independence.

## Contribution

It introduces a novel multiscale statistic for quantile segmentation that controls the probability of correct segment detection and achieves minimax optimal estimation rates.

## Key findings

- Method reliably detects quantile changes in simulations.
- Provides asymptotic confidence bands for quantile functions.
- Effective in genetic sequencing and ion channel data analysis.

## Abstract

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.