# The exponential distribution analog of the Grubbs--Weaver method

## TL;DR

This paper establishes a new optimal estimator for the exponential distribution's scale parameter, analogous to the Grubbs-Weaver method for normal distributions, and proves the optimality of a 'rule of fours' for subsample size.

## Contribution

It introduces and rigorously proves an exponential distribution analog of the Grubbs-Weaver estimator, replacing the 'rule of eights' with a 'rule of fours' for optimality.

## Key findings

- The 'rule of fours' is optimal for the exponential distribution.
- The estimator minimizes variance among unbiased estimators.
- The result is rigorously proven for all sample sizes.

## Abstract

Grubbs and Weaver (JASA 42 (1947) 224--241) suggest a minimum-variance unbiased estimator for the population standard deviation of a normal random variable, where a random sample is drawn and a weighted sum of the ranges of subsamples is calculated. The optimal choice involves using as many subsamples of size eight as possible. They verified their results numerically for samples of size up to 100, and conjectured that their "rule of eights" is valid for all sample sizes. Here we examine the analogous problem where the underlying distribution is exponential and find that a "rule of fours" yields optimality and prove the result rigorously.