Comparison of multivariate distributions using quantile-quantile plots and related tests

TL;DR

This paper introduces a multivariate extension of the quantile-quantile plot using spatial quantiles, along with related statistical tests, to compare and assess multivariate distributions effectively, especially in high-dimensional data.

Contribution

It develops a novel multivariate Q-Q plot based on spatial quantiles and proposes related statistical tests with studied asymptotic properties.

Findings

The multivariate Q-Q plot is effective for high-dimensional data.

Proposed tests outperform some existing methods in certain scenarios.

The methods are validated on real and simulated datasets.

Abstract

The univariate quantile-quantile (Q-Q) plot is a well-known graphical tool for examining whether two data sets are generated from the same distribution or not. It is also used to determine how well a specified probability distribution fits a given sample. In this article, we develop and study a multivariate version of the Q-Q plot based on the spatial quantile. The usefulness of the proposed graphical device is illustrated on different real and simulated data, some of which have fairly large dimensions. We also develop certain statistical tests that are related to the proposed multivariate Q-Q plot and study their asymptotic properties. The performance of those tests are compared with that of some other well-known tests for multivariate distributions available in the literature.