Can we define a best estimator in simple 1-D cases ?

TL;DR

This paper investigates the differences among common estimators for a single scale parameter in small-sample scenarios and shows how a logarithmic transformation can unify their results.

Contribution

It introduces a transformation approach that makes different estimators equivalent in the absence of prior information, clarifying estimator selection in simple 1-D cases.

Findings

Transforming the scale parameter to a location parameter via logarithms aligns estimators.

Estimator differences diminish when data are log-transformed.

The approach simplifies estimator comparison in small-sample contexts.

Abstract

With a small number of measurements, the three most well known estimators of a single parameter give very different results when estimating a scale parameter. A transformation of this scale parameter to a location parameter by using logarithms of the data rends the three estimators equivalent in the absence of any a priori information.