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

F. Lombard; I.J.H. Visagie

arxiv: 1906.12106 · v1 · pith:OJCBSEPGnew · submitted 2019-06-28 · 📊 stat.ME · stat.AP

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

I.J.H. Visagie , F. Lombard This is my paper

Pith reviewed 2026-05-25 14:04 UTC · model grok-4.3

classification 📊 stat.ME stat.AP

keywords conditional distributionmean of two closestnormal distributionLaplace distributionreplicate measurementsthreshold ruleestimation procedure

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The pith

The conditional distribution of the average of the third measurement and the closest of the first two differs markedly under normal versus Laplace assumptions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies an estimation procedure used in chemical analyses: take two measurements, use their average if they agree closely enough, but if they differ by more than a fixed threshold obtain a third measurement and average it with whichever of the first two is closer. The focus is the distribution of this final estimate conditional on the threshold being crossed. A reader would care because the procedure is applied in practice to combine replicates, so its sampling behavior under realistic error models matters for inference. The central result is that this conditional distribution takes a markedly different form when the measurements are modeled as normal compared with when they are modeled as Laplace.

Core claim

Conditional on the first two observations differing by more than a preset threshold, the distribution of the average formed by the third observation and the closer of the first two is markedly different when the three observations are independent and identically distributed normal random variables than when they are independent and identically distributed Laplace random variables.

What carries the argument

The conditional distribution of the mean of the two closest observations given that the initial pair exceeds the difference threshold.

If this is right

Inference procedures that treat the estimator as having the same properties regardless of error distribution will be incorrect in this conditional setting.
The variance and shape of the estimator depend on whether the underlying distribution is taken to be normal or Laplace.
Any attempt to combine the estimate with other data or to form intervals must incorporate the specific conditional law that arises from the triggering rule.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result supplies a concrete reason to test the error distribution before relying on the numerical properties of this particular averaging rule.
If real measurement errors are closer to one model than the other, the choice of model changes the effective precision of the final estimate.

Load-bearing premise

The three observations are independent and identically distributed from either a normal distribution or a Laplace distribution.

What would settle it

Generate or collect replicate triples from a known normal distribution and from a known Laplace distribution, apply the rule that triggers the third measurement, and check whether the empirical conditional distributions of the resulting estimates differ in the way the paper derives.

Figures

Figures reproduced from arXiv: 1906.12106 by F. Lombard, I.J.H. Visagie.

**Figure 1.** Figure 1: Density of µb given that |X1 − X2| > r under the normal (dashed line) and Laplace (solid line) assumptions. -4 -3 -2 -1 0 1 2 3 4 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Densities as functions of x [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: Standardised normal (dashed line) and Laplace (so [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: g (x, α) (solid line) and g (−x, α) (dashed line) under the normal distribution. -4 -3 -2 -1 0 1 2 3 4 x 0 1 2 3 4 5 6 7 8 g(x,alpha) and g(-x,alpha) under the Laplace model 10-3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: g (x, α) (solid line) and g (−x, α) (dashed line) under the Laplace distribution. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Exceedance probabilities for the normal (dashed l [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Difference in observed ash contents for 199 batches [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives explicit conditional distributions for this specific 'two-plus-one' estimator under normal and Laplace errors and shows they differ, which is a clean but narrow extension of order-statistic work.

read the letter

The main point is that the authors work out the conditional distribution of the estimator (average of the third observation and whichever of the first two is closest to it) given that the initial pair differs by more than a fixed threshold. They do this separately under iid normal errors and iid Laplace errors, and the two conditional laws are not the same. That is the actual new content: explicit expressions for this exact procedure rather than a restatement of general order statistics. The derivations follow directly from the stated assumptions without circularity or fitted parameters, so the central claim holds up on its own terms. The difference is expected once you think about tail behavior and the conditioning event, but spelling it out for this estimator is still useful for the chemical-assay setting they describe. The work is narrow in scope. It will mainly interest people who already use or study this particular duplicate-or-triplicate rule in metrology and want the exact conditional law for uncertainty calculations. Broader statistical theory is not moved. No obvious gaps in the logic appear from the abstract and stress-test note, though without the full derivations one cannot check the algebra in detail. A reader who needs these distributions for applied work will find them here; someone looking for general results on conditional order statistics will not. The paper is coherent and engages the literature on its own terms, so it should go to referees rather than be desk-rejected.

Referee Report

0 major / 3 minor

Summary. The manuscript examines an estimation procedure for a true value using up to three i.i.d. observations drawn from either a normal or Laplace distribution. Two initial observations are taken; if their absolute difference is below a fixed threshold their average is the estimate, otherwise a third observation is obtained and the estimate is defined as the average of the third observation and whichever of the first two is closer to it. The paper focuses on the conditional distribution of this estimator given that the initial pair exceeds the threshold and reports that these conditional distributions differ markedly between the normal and Laplace cases.

Significance. If the derivations hold, the result provides a concrete illustration of how the tail behavior of the parent distribution affects the conditional law of this practical 'two-plus-one' estimator after conditioning on a large initial discrepancy. This is relevant to applied settings such as chemical assay where such rules are used and where normality is often assumed without verification; the explicit normal-versus-Laplace comparison quantifies the sensitivity and supplies a benchmark for robustness checks.

minor comments (3)

The abstract states the main finding but supplies no indication of the analytic or numerical methods used to obtain the conditional distributions; a brief sentence in the abstract or a pointer to the relevant section would improve accessibility.
Notation for the threshold and the ordering of the three observations should be introduced once and used consistently; several places appear to switch between |X1-X2| and the generic 'difference' without cross-reference.
Figure captions (if present) should explicitly state whether the displayed densities are exact, simulated, or approximated, and should include the value of the threshold used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for recognizing its potential relevance to applied settings such as chemical assays, and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives the conditional distribution of the estimator (average of third observation and closest of first two, given |X1-X2| exceeds threshold) directly from the explicit iid normal or Laplace assumptions stated in the abstract and methods. No equations reduce to fitted parameters renamed as predictions, no self-citations are load-bearing for the central claim, and the derivations are self-contained mathematical consequences of the model assumptions without self-definition or imported uniqueness theorems. The observed difference between normal and Laplace cases follows from their distinct tail behaviors under the conditioning event and does not rely on any internal reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption of i.i.d. normal or Laplace errors and independence of the three measurements; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The three observations are independent and identically distributed according to either a normal or Laplace distribution.
This assumption is invoked to derive the conditional distributions when the initial pair differs by more than the threshold.

pith-pipeline@v0.9.0 · 5719 in / 1134 out tokens · 35207 ms · 2026-05-25T14:04:54.116033+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Keynes, J.M. (1911). The principal averages and the laws of error which lead to them. Journal of the Royal Statistical Society, 74, 322-331

work page 1911
[2]

Lieblein, J. (1952). Properties of certain statistics i nvolving the closest pair in a sam- ple of three observations , Journal of Research of the National Bureau of Standards, 48 (3), 255-268

work page 1952
[3]

MATLAB Release 2018b, The MathWorks, Inc., Natick, Mass achusetts, United States

work page
[4]

Seth, G.R. (1950). On the distribution of the two closest among a set of three obser- vations. The Annals of Mathematical Statistics, 21(2), 298-301

work page 1950
[5]

Wilson, E.B. (1923). First and second laws of error. Journal of the American Statis- tical Association, 18, 841-851. 4 Appendix 1: Derivation of (5) Let X1 and X2 denote the ﬁrst two observations and let X3 denote the third sample observation. Given x and a small δ > 0, let dx denote the interval (x − δ, x + δ). Then P [X ∈ dx ||X1 − X2| > r ] = P [X ∈ dx...

work page 1923

[1] [1]

Keynes, J.M. (1911). The principal averages and the laws of error which lead to them. Journal of the Royal Statistical Society, 74, 322-331

work page 1911

[2] [2]

Lieblein, J. (1952). Properties of certain statistics i nvolving the closest pair in a sam- ple of three observations , Journal of Research of the National Bureau of Standards, 48 (3), 255-268

work page 1952

[3] [3]

MATLAB Release 2018b, The MathWorks, Inc., Natick, Mass achusetts, United States

work page

[4] [4]

Seth, G.R. (1950). On the distribution of the two closest among a set of three obser- vations. The Annals of Mathematical Statistics, 21(2), 298-301

work page 1950

[5] [5]

Wilson, E.B. (1923). First and second laws of error. Journal of the American Statis- tical Association, 18, 841-851. 4 Appendix 1: Derivation of (5) Let X1 and X2 denote the ﬁrst two observations and let X3 denote the third sample observation. Given x and a small δ > 0, let dx denote the interval (x − δ, x + δ). Then P [X ∈ dx ||X1 − X2| > r ] = P [X ∈ dx...

work page 1923