Higher-order asymptotic corrections and their application to the Gamma Variance Model

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).