Depth based inference on conditional distribution with infinite dimensional data

TL;DR

This paper introduces nonparametric depth-based methods for inference on conditional distribution characteristics like dispersion and skewness in complex, high-dimensional, and functional data settings, with proven consistency and practical tests.

Contribution

It develops novel depth-based nonparametric procedures for measuring and testing conditional dispersion and skewness in multivariate and functional data, including asymptotic properties.

Findings

Methods are asymptotically consistent.

Tests for heteroscedasticity and skewness are effective.

Application to real data demonstrates practical utility.

Abstract

We develop inference and testing procedures for conditional dispersion and skewness in a nonparametric regression setup based on statistical depth functions. The methods developed can be applied in situations, where the response is multivariate and the covariate is a random element in a metric space. This includes regression with functional covariate as a special case. We construct measures of the center, the spread and the skewness of the conditional distribution of the response given the covariate using depth based nonparametric regression procedures. We establish the asymptotic consistency of those measures and develop a test for heteroscedasticity and a test for conditional skewness. We present level and power study for the tests in several simulated models. The usefulness of the methodology is also demonstrated in a real dataset. In that dataset, our responses are the nutritional contents of different meat samples measured by their protein, fat and moisture contents, and the functional covariate is the absorbance spectra of the meat samples.