 Thermal-product dependence beyond covariance

Thermal-product dependence beyond covariance is the influence of higher joint moments on uncertainty in a nonlinear thermal expansion correction. Two input models can have identical means, variances and covariance yet predict different uncertainty in a product. A thermal-product input-distribution check must therefore examine how the temperature offset and expansion-coefficient error occur together.

Let $a={\mu }_a+A$ and $\theta ={\mu }_{\theta }+T$, where the centered variables have expected values zero, variances $s_a^2,s_{\theta }^2$ and covariance $c=\mathbb{E}[AT]$. Direct expansion givesThe third joint moments and the mixed fourth moment of thermal inputs are extra information. For centered inputs, only the last two terms remain. A first-order uncertainty propagation rule at zero input means would miss the product entirely; the GUM explicitly requires higher-order treatment when nonlinearity matters.^[ref]

An original counterexample keeps even the marginal shapes fixed. Let $Z$ have a standard normal distribution and let $S$ independently take values $+1$ and $-1$ with equal probabilities. Define $A=s_{a}Z$ and $T=s_{\theta }SZ$. Both have centered normal distributions, the quoted variances and zero covariance. They are not independent: their absolute standardized values are identical.

Their product is $s_a{s}_{\theta }S{Z}^2$. Its expected value is zero, while its variance is $3s_a^2{s}_{\theta }^2$, because $\mathbb{E}[Z^4]=3$. Independent centered normal inputs with the same individual variances instead give product variance $s_a^2{s}_{\theta }^2$. The joint model, rather than a marginal histogram or correlation coefficient alone, determines the difference.

For an illustrative length correction $d=La\theta$ with zero means, $L=100\mathrm{mm}$, $s_a=10^{-6}{\mathrm{K}}^{-1}$ and $s_{\theta }=0.1\mathrm{K}$, these models give standard deviations $\sqrt{3}\times 10\approx 17.32$ nm and 10 nm. This constructed dependence is a mathematical test case, not a proposed universal thermometer model.

In a physical gauge block study, track paired temperature and coefficient information, shared thermometer corrections and any procedure that estimates one input using the other. NIST's thermal correction explicitly combines block length, expansion coefficient and temperature difference.^[ref] Preserve that dependence when generating joint draws. JCGM 101 distinguishes model formulation from propagation and gives a multivariate Gaussian input model as a specific choice, not a consequence of normal marginals alone.^[ref] More simulation trials reduce numerical error conditional on the chosen model; they cannot identify unmeasured joint moments.