  We present improved methods for calculating confidence intervals and
$p$-values in situations where standard asymptotic approaches fail due to small
sample sizes. We apply these techniques to a specific class of statistical
model that can incorporate uncertainties in parameters that themselves
represent uncertainties (informally, "errors on errors") called the Gamma
Variance Model. This model contains fixed parameters, generically called
$\varepsilon$, that represent the relative uncertainties in estimates of
standard deviations of Gaussian distributed measurements. If the $\varepsilon$
parameters are small, one can construct confidence intervals and $p$-values
using standard asymptotic methods. This is formally similar to the familiar
situation of a large data sample, in which estimators for all adjustable
parameters have Gaussian distributions. Here we address the important case
where the $\varepsilon$ parameters are not small and as a consequence the
asymptotic distributions do not represent a good approximation. We investigate
improved test statistics based on the technology of higher-order asymptotics
($p^*$ approximation and Bartlett correction).
