Alessandra Rosalba Brazzale

arXiv2026

Likelihood-Based Confidence Curves for Bounded Parameters

When a parameter is bounded, the likelihood ratio statistic does not generally follow the chi-square limit distribution of Wilks' theorem. In fact, its asymptotic null distribution depends on the distance from the null value to the boundary, interpolating between the chi-bar-square law on the boundary and the chi-square law in the deep interior of the parameter space. We derive this distribution and use it to construct confidence curves, which are nested collections of two-sided confidence intervals or regions, for bounded parameters. We extend the existing result for the scalar parameter to a two-dimensional parameter with one or both components bounded. As the bivariate distributions are integrals of normal distributions, which are not reducible to standard statistical functions, we also give a simpler approximated distribution obtained by moment-matching. We illustrate the methods on a canonical analysis from high-energy physics, namely the Poisson signal-plus-background experiment.

stat.ME↗

arXiv2024

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).

physics.data-an↗