= Ellipsoidal admissibility test for overlap data
An <ellipsoidal admissibility test for overlap data> checks whether every point of a specified joint calibration <ellipsoid> permits finite Gaussian radius inversion. For the <inversion of a symmetric Gaussian overlap log-ratio>, write $x=(D,h)^{\mathsf T}$ with $D$ the measured logarithmic ratio and $h$ the positive radius-setting parameter. The admissible set is $D-4h<0$ and $-D-4h<0$. Together these strict inequalities already imply $h>0$.
Suppose the joint set is
$$
\mathcal E=\{\mu+Ay:\|y\|\le k\},\qquad \Sigma=AA^{\mathsf T},\quad k\ge0.
$$
For nonsingular $\Sigma$ this is equivalently $(x-\mu)^{\mathsf T}\Sigma^{-1}(x-\mu)\le k^2$. The parameterized form also accommodates a singular <covariance matrix> or a lower-dimensional deterministic set. Its <support function> in direction $l$ is
$$
\max_{x\in\mathcal E}l^{\mathsf T}x=l^{\mathsf T}\mu+k\sqrt{l^{\mathsf T}\Sigma l}.
$$
Indeed, $l^{\mathsf T}Ay=(A^{\mathsf T}l)^{\mathsf T}y$, so the <Cauchy-Schwarz inequality> gives the bound, attained by a parallel $y$ when $A^{\mathsf T}l\ne0$.
Let $\mu=(D_0,h_0)^{\mathsf T}$, marginal scales $s_D,s_h$, and off-diagonal entry $c_{Dh}$. Substituting $l_+=(1,-4)^{\mathsf T}$ and $l_-=(-1,-4)^{\mathsf T}$ gives two exact tests:
$$
D_0-4h_0+k\sqrt{s_D^2+16s_h^2-8c_{Dh}}<0,
$$
$$
-D_0-4h_0+k\sqrt{s_D^2+16s_h^2+8c_{Dh}}<0.
$$
Both must hold. Equality means that the closed joint set touches a boundary where no finite positive radius ratio exists; testing only the center would miss that point.
For an original example, take $D_0=2.4$, $h_0=0.7$, $s_D=0.1$, $s_h=0.03$ and $k=2$. Zero <covariance> gives upper-boundary excursion $2\sqrt{0.0244}\approx0.3124$, smaller than the central margin $0.4$. The lower-boundary test also passes. With <correlation coefficient> $-0.8$, however, $c_{Dh}=-0.0024$ and the upper excursion becomes $2\sqrt{0.0436}\approx0.4176$: the joint set crosses the forbidden region despite identical marginal scales.
The offending point need not be found by grid search. When $l^{\mathsf T}\Sigma l>0$, the maximizing point is
$$
x_* =\mu+\frac{k\Sigma l}{\sqrt{l^{\mathsf T}\Sigma l}}.
$$
For the correlated example and $l=l_+$, this gives approximately $(D,h)=(2.5877,0.6425)$, with $D-4h\approx0.0176>0$. It is a concrete incompatible calibration pair.
The <ellipsoid> here is a stated joint set. Interpreting it as a <confidence region> requires a separate sampling or uncertainty model; choosing $k=2$ alone supplies no universal coverage probability. JCGM 101 distinguishes assignment of joint input distributions from propagation and summaries.<Propagation of distributions using a Monte Carlo method by JCGM>[§§5.1–5.3 and 5.10]{ref} If the set fails, report the incompatible region and reconsider the input model or measurement. Clipping it before inversion changes the set being propagated.
Back to article page
