# On the conditional distribution of the mean of the two closest among a set of three observations

## TL;DR

This paper investigates the statistical properties of a new estimation method for chemical assay values, which adaptively combines two or three measurements based on their differences, analyzing its distribution under normal and Laplace assumptions.

## Contribution

It introduces a novel adaptive estimator for chemical measurements and derives its conditional distribution under different distributional assumptions.

## Key findings

- Conditional distributions differ significantly between normal and Laplace models.
- The proposed method improves estimation accuracy when initial measurements differ greatly.
- Analytical expressions for the estimator's distribution are provided.

## Abstract

Chemical analyses of raw materials are often repeated in duplicate or triplicate. The assay values obtained are then combined using a predetermined formula to obtain an estimate of the true value of the material of interest. When duplicate observations are obtained, their average typically serves as an estimate of the true value. On the other hand, the "best of three" method involves taking three measurements and using the average of the two closest ones as estimate of the true value. In this paper, we consider another method which potentially involves three measurements. Initially two measurements are obtained and if their difference is sufficiently small, their average is taken as estimate of the true value. However, if the difference is too large then a third independent measurement is obtained. The estimator is then defined as the average between the third observation and the one among the first two which is closest to it. Our focus in the paper is the conditional distribution of the estimate in cases where the initial difference is too large. We find that the conditional distributions are markedly different under the assumption of a normal distribution and a Laplace distribution.