• Bernoulli
  • Volume 25, Number 1 (2019), 395-423.

Nonparametric depth and quantile regression for functional data

Joydeep Chowdhury and Probal Chaudhuri

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We investigate nonparametric regression methods based on spatial depth and quantiles when the response and the covariate are both functions. As in classical quantile regression for finite dimensional data, regression techniques developed here provide insight into the influence of the functional covariate on different parts, like the center as well as the tails, of the conditional distribution of the functional response. Depth and quantile based nonparametric regression methods are useful to detect heteroscedasticity in functional regression. We derive the asymptotic behavior of the nonparametric depth and quantile regression estimates, which depend on the small ball probabilities in the covariate space. Our nonparametric regression procedures are used to analyze a dataset about the influence of per capita GDP on saving rates for 125 countries, and another dataset on the effects of per capita net disposable income on the sale of cigarettes in some states in the US.

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Bernoulli, Volume 25, Number 1 (2019), 395-423.

Received: June 2016
Revised: September 2017
First available in Project Euclid: 12 December 2018

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Bahadur representation conditional spread convergence rates maximal depth set spatial depth spatial quantile


Chowdhury, Joydeep; Chaudhuri, Probal. Nonparametric depth and quantile regression for functional data. Bernoulli 25 (2019), no. 1, 395--423. doi:10.3150/17-BEJ991.

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Supplemental materials

  • Supplement to “Nonparametric depth and quantile regression for functional data”. We present here the plots of confidence sets for conditional spatial medians in the examples considered in the paper. In addition, some mathematical details required for the proofs of the theorems are provided.