Unbiasedness and efficiency of non-parametric and UMVUE estimators of the probabilistic index and related statistics

Abstract

In reliability theory, diagnostic accuracy, and clinical trials, the quantity [Formula: see text], also known as the Probabilistic Index (PI), is a common treatment effect measure when comparing two groups of observations. The quantity [Formula: see text], a linear transformation of PI known as the net benefit, has also been advocated as an intuitively appealing treatment effect measure. Parametric estimation of PI has received a lot of attention in the past 40 years, with the formulation of the Uniformly Minimum-Variance Unbiased Estimator (UMVUE) for many distributions. However, the non-parametric Mann–Whitney estimator of the PI is also known to be UMVUE in some situations. To understand this seeming contradiction, in this paper a systematic comparison is performed between the non-parametric estimator for the PI and parametric UMVUE estimators in various settings. We show that the Mann–Whitney estimator is always an unbiased estimator of the PI with univariate, completely observed data, while the parametric UMVUE is not when the distribution is misspecified. Additionally, the Mann–Whitney estimator is the UMVUE when observations belong to an unrestricted family. When observations come from a more restrictive family of distributions, the loss in efficiency for the non-parametric estimator is limited in realistic clinical scenarios. In conclusion, the Mann–Whitney estimator is simple to use and is a reliable estimator for the PI and net benefit in realistic clinical scenarios.