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A leave-one-out interlaboratory consensus estimates a benchmark for laboratory $i$ from the other participants. For a fixed weighted mean with positive coefficients $w_j$ and at least two results, defineThe omitted result does not enter its own benchmark. This removes the direct participant influence on a consensus assigned value for the stated fixed-weight estimator. It does not establish independence when laboratories share a systematic error. The proficiency-testing guide distinguishes participant consensus from expert or independent reference assignment; the leave-one-out construction here is a mathematical comparison, not a claim that the guide requires this scoring procedure.^[ref]

With equal weights and original results $(9,10,11)$, the omitted-result arithmetic means are $(10.5,10,9.5)$, giving residuals $(-1.5,0,1.5)$. Residuals from the full arithmetic mean are $(-1,0,1)$. In general,The larger magnitudes reflect a changed comparator rather than a changed laboratory result.

The joint dependence of leave-one-out consensus residuals matters when interpreting the collection of scores. Suppose the $X_i$ are independent random variables with a common mean and finite common variance ${\sigma }^2$. A result is independent of the arithmetic mean of the other results, soFor different participants, however, their leave-one-out benchmarks overlap. Linearity of covariance givesThus independent laboratory observations do not give independent residual scores. The zero-sum identity makes their covariance matrix a singular matrix. For $n=4$ and $\sigma =2$ units, each residual has standard deviation $\sqrt{16/3}\approx 2.31$ units, and each distinct pair has correlation coefficient $-1/3$. No normal distribution assumption is needed for these moment identities.

A leave-one-out inverse-variance weighted consensus applies an inverse-variance weighted mean to the remaining participants when their standard uncertainties are known. Under independent unbiased observations with weights $w_j=u_j^{-2}$, put $W={\sum }_j{w}_j$. ThenThese are uncertainties under that observation model; the resulting quantity should not automatically be called a proficiency-test z-score with a fixed assessment scale. Computing $({\sum }_j{w}_j{x}_j-w_i{x}_i)/(W-w_i)$ is algebraically convenient, but direct summation of the remaining terms can avoid cancellation when one participant dominates the total weight.

If a robust estimator has weights depending on the full data set, merely deleting one weighted numerator term need not reproduce the estimator fitted without that participant. The estimator must be recomputed according to its specified rule. Finally, a common additive calibration error remains undetectable in these residuals: replacing every $X_i$ by $X_i+B$ leaves all $D_i$ unchanged. Shared bias in an interlaboratory consensus therefore remains a separate question from self-influence and the variance of the residuals.

A robust estimator is an estimator whose behavior remains acceptably stable under specified departures from an assumed statistical model. “Stable” must identify a criterion, such as limited bias under contamination, bounded response to a single extreme observation, or resistance to replacing a fraction of the sample. It does not mean that the estimator is accurate under every possible probability distribution.

For example, consider data $(-1,0,1,M)$ with $M\ge 1$. The sample mean is $M/4$, whereas the sample median, using the midpoint convention for an even-sized sample, is $1/2$. Sending $M$ to infinity makes the sample mean diverge without changing that sample median. The example concerns location estimation: the population median and population mean need not coincide outside a symmetric reference model.

One formal model is Huber contamination:where $F$ is a reference probability distribution, $G$ is an unspecified contaminating probability distribution, and $\epsilon$ is its proportion. A robustness claim can bound an estimator’s worst bias over allowed $G$, rather than only calculating its variance when the reference model is exactly true. Peter J. Huber’s original location paper treats this tradeoff systematically.^[ref]

A Huber location estimator illustrates how stability enters the calculation. With a specified positive scale $s$, solveLarge residuals have bounded scores, so one increasingly extreme point cannot contribute an arbitrarily large term to this equation. By contrast, the squared-error score grows linearly with the residual and yields the sample mean. A practical analysis must also state how $s$ is obtained: a scale estimate that itself explodes under outliers can undermine the intended protection.

Different criteria answer different questions. An influence function studies an infinitesimal perturbation of a distribution, while a breakdown point concerns a finite amount of sample corruption. A robust estimator can sacrifice some model-specific statistical efficiency to improve performance away from that model, but the amount depends on the chosen tuning and departure. Reporting the target parameter, reference model and robustness criterion makes the claim testable.