Leave-one-out inverse-variance weighted consensus (${\hat{\theta }}_{-i}$)

A leave-one-out inverse-variance weighted consensus applies an inverse-variance weighted mean to all laboratories except the one being assessed. Suppose their results $X_j$ are independent random variables with common expected value $\theta$ and known positive variances $u_j^2$. For laboratory $i$, defineUnder this model it is the best linear unbiased estimator of $\theta$ from the remaining results. The omitted observation does not determine its own leave-one-out interlaboratory consensus value.

The minimum-variance property follows directly from the Cauchy-Schwarz inequality. For any fixed coefficients $c_j$ satisfying ${\sum }_{j\ne i}{c}_j=1$, the linear unbiased estimator $T={\sum }_{j\ne i}{c}_j{X}_j$ hasEquality holds precisely for $c_j=u_j^{-2}/W_{-i}$. ThereforeThis is a comparison among linear unbiased estimators under the stated independent-error model; it requires no normal distribution assumption.

For an original example, assess a laboratory reporting $x_i=12$ with standard uncertainty $u_i=0.5$. The other laboratories report $(10,12,8)$ with respective standard uncertainties $(1,2,4)$, all in the same units. Their total precision is $W_{-i}=1+1/4+1/16=21/16$, givingThe held-out difference is $D_i=12/7\simeq 1.714$, withThe denominator includes the benchmark's uncertainty even though $X_i$ was excluded. This uncertainty-normalized difference and a proficiency-test z-score based on a predetermined assessment scale answer different questions.

Shared bias in an interlaboratory consensus changes the uncertainty model. Suppose instead $X_j=\theta +B+{\epsilon }_j$, where the common reference contribution $B$ has variance $u_B^2$ and the mutually independent residuals, independent also of $B$, have variances $s_j^2$. Coefficients summing to one preserve $B$ in the benchmark. Its variance is $u_B^2+\sum c_j^2{s}_j^2$, so the minimizing coefficients use $s_j^{-2}$, andUsing the full marginal uncertainties $\sqrt{u_B^2+s_j^2}$ as if they were independent changes the coefficients and loses the shared-reference structure. The propagation of uncertainty retains the covariance of dependent inputs.^[ref]

If the held-out laboratory shares the same $B$, that contribution cancels in $D_i$, while it remains in the benchmark's absolute measurement uncertainty. Precision of a difference and knowledge of an absolute value are distinct properties. Estimated, result-dependent uncertainties or additional correlations require analysis of the actual estimator; the fixed known-variance proof does not establish optimality for those cases.