Nonparametric inference for $P(X<Y)$ with paired variables

TL;DR

This paper introduces new nonparametric estimators for the probability that one variable is less than another in paired, possibly dependent data, emphasizing the importance of accounting for pairing and dependence.

Contribution

It develops two classes of nonparametric estimators for $P(X<Y)$ in paired data, including methods for confidence intervals and convergence analysis.

Findings

Estimators perform well in simulations and real data.

Ignoring pairing can lead to incorrect conclusions.

Bootstrap methods effectively provide confidence intervals.

Abstract

We propose two classes of nonparametric point estimators of $\theta=P(X<Y)$ in the case where $(X,Y)$ are paired, possibly dependent, absolutely continuous random variables. The proposed estimators are based on nonparametric estimators of the joint density of $(X,Y)$ and the distribution function of $Z=Y - X$. We explore the use of several density and distribution function estimators and characterise the convergence of the resulting estimators of $\theta$. We consider the use of bootstrap methods to obtain confidence intervals. The performance of these estimators is illustrated using simulated and real data. These examples show that not accounting for pairing and dependence may lead to erroneous conclusions about the relationship between $X$ and $Y$.