= Quadratic Gaussian overlap correction
A \b[quadratic Gaussian overlap correction] models an estimated logarithmic variance as a nonnegative quadratic function of Gaussian calibration residuals. It gives an exact mean for a corrected overlap while retaining the dependence between the reading and the variance estimate. The model is useful for testing <uncertainty propagation> software and understanding which shared residuals matter.
Let $Z\sim N(0,I_n)$, $d=a^{\mathsf T}Z$ and $\widehat v=c+Z^{\mathsf T}BZ$, with $B$ a symmetric <positive semidefinite matrix> and $c\ge0$. For $\widetilde\gamma=\gamma\exp(d-\widehat v/2)$, the <joint exponential moment in overlap correction> gives
$$
\frac{\mathbb E[\widetilde\gamma]}\gamma
=\frac{e^{-c/2}}{\sqrt{\det(I_n+B)}}
\exp\!\left[\frac12a^{\mathsf T}(I_n+B)^{-1}a\right].
$$
The eigenvalues of $B$ describe how strongly each calibration direction enters the variance estimate, while the components of $a$ describe how those same directions enter the reading. <Eigenmode alignment in overlap variance correction> therefore affects the bias even when the variance estimate's marginal distribution is unchanged.
An original eigenvector calculation writes $B=Q\operatorname{diag}(\lambda_k)Q^{\mathsf T}$ and $\alpha=Q^{\mathsf T}a$. Taking logarithms gives
$$
\log\frac{\mathbb E[\widetilde\gamma]}\gamma
=-\frac c2-\frac12\sum_k\log(1+\lambda_k)
+\frac12\sum_k\frac{\alpha_k^2}{1+\lambda_k}.
$$
This form avoids taking a determinant and a potentially large exponential separately. It also displays which calibration directions overlap the readout direction. Unbiased variance estimation requires $c+\sum_k\lambda_k=\|a\|^2$; it does not force the displayed logarithm to vanish.
For a stipulated two-direction example, take $a=(1,0)^{\mathsf T}$, $c=0.9$ and one nonzero eigenvalue 0.1. If that eigenvalue lies along the reading direction, $B=\operatorname{diag}(0.1,0)$ and the mean ratio is about 0.9578. Rotating it into the independent direction, $B=\operatorname{diag}(0,0.1)$, gives about 1.00235. Both estimates have the form $0.9+0.1Z_1^2$ in distribution, and both have mean one. Only their joint relationship to $d$ changes.
A small-$B$ expansion, with $c=\|a\|^2-\operatorname{tr}B\ge0$, makes the distinction sharper:
$$
\log\frac{\mathbb E[\widetilde\gamma]}\gamma
=-\frac12a^{\mathsf T}Ba+\frac14\operatorname{tr}(B^2)
+\frac12a^{\mathsf T}B^2a+O(\|B\|^3).
$$
The shared-direction term can appear at first order; the independent-direction contribution begins differently. All examples are probability calculations, not measured receiver performance.
In an actual analysis, build $a$ and $B$ from the stated measurement model and reused calibration records. JCGM 101 distinguishes input-distribution assignment from propagation and provides multivariate Gaussian inputs as one particular joint model.<Propagation of distributions using a Monte Carlo method by JCGM>[§§5.1–5.2 and 6.4.8]{ref} Agreement of a <Monte Carlo method> with the exact formula checks implementation conditional on those inputs. It does not establish that Gaussian residuals or a quadratic variance estimator describe the physical experiment.
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