On continuous distribution functions, minimax and best invariant estimators, and integrated balanced loss functions

TL;DR

This paper develops optimal invariant estimators for continuous distribution functions using minimax and balanced loss functions, with applications to sampling methods and medical data analysis.

Contribution

It introduces integrated balanced loss functions and derives their optimal estimators, extending risk analysis and establishing minimaxity and dominance results.

Findings

Best invariant estimators are minimax for certain loss functions.

Integrated balanced loss functions effectively combine distance and target proximity.

Applications include sampling methods and bilirubin level analysis in jaundiced babies.

Abstract

We consider the problem of estimating a continuous distribution function $F$, as well as meaningful functions $\tau(F)$ under a large class of loss functions. We obtain best invariant estimators and establish their minimaxity for H\"{o}lder continuous $\tau$'s and strict bowl-shaped losses with a bounded derivative. We also introduce and motivate the use of integrated balanced loss functions which combine the criteria of an integrated distance between a decision $d$ and $F$, with the proximity of $d$ with a target estimator $d_0$. Moreover, we show how the risk analysis of procedures under such an integrated balanced loss relates to a dual risk analysis under an "unbalanced" loss, and we derive best invariant estimators, minimax estimators, risk comparisons, dominance and inadmissibility results. Finally, we expand on various illustrations and applications relative to maxima-nomination sampling, median-nomination sampling, and a case study related to bilirubin levels in the blood of babies suffering from jaundice.