On frequentist confidence intervals in a non-Gaussian regime

TL;DR

The paper assesses how different frequentist confidence-interval constructions perform when MCMC posteriors are non-Gaussian in a two-parameter $\Lambda$CDM framework using cosmic chronometer data and DESI mocks. It compares graphical profile likelihoods (Wilks and integration) with the Feldman-Cousins (FC) Neyman belt, all driven by the same MCMC chain to ensure a fair baseline. In near-Gaussian regimes the methods align within about 10%, but non-Gaussianity and parameter boundaries lead to noticeable discrepancies, with FC providing robust coverage and sometimes tighter intervals. A key finding is a ~2$\sigma$ shift in $\Omega_m$ between low- and high-redshift CC subsets when mocks are fitted to all parameters, while fixing parameters artificially tightens intervals and can mask non-Gaussian features. The results emphasize cautious use of Gaussian approximations in cosmology, advocate FC/Neyman approaches in non-Gaussian settings, and highlight that the number of fitted parameters strongly shapes interval widths and inferred tensions.

Abstract

We study frequentist confidence intervals based on graphical profile likelihoods (Wilks' theorem, likelihood integration), and the Feldman-Cousins (FC) prescription, a generalisation of the Neyman belt construction, in a setting with non-Gaussian Markov chain Monte Carlo (MCMC) posteriors. Our simplified setting allows us to recycle the MCMC chain as an input in all methods, including mock simulations underlying the FC approach. We find all methods agree to within $10 \%$ in the close to Gaussian regime, but extending methods beyond their regime of validity leads to greater discrepancies. Importantly, we recover a $\sim 2 σ$ shift in cosmological parameters between low and high redshift cosmic chronometer data with the FC method, but only when one fits all parameters back to the mocks. We observe that fixing parameters, a common approach in the literature, risks underestimating confidence intervals.

Figures (16)

• Figure 1: 33 OHD constraints from the CC program.
• Figure 2: 29 OHD constraints from DESI forecast with canonical $\Lambda$CDM cosmology.
• Figure 3: MCMC posteriors for CC OHD with 5 low redshift cuts $z > z_{\textrm{min}}$. 2D posteriors trace curves of constant $H_0 \sqrt{\Omega_m}$ at higher redshifts as this is the only parameter combination well constrained by the data. The marginalised 1D posteriors become steadily less Gaussian as effective redshift is increased. $H_0$ is in units of km/s/Mpc throughout this work.
• Figure 4: Same as Fig. \ref{['fig:CC_posteriors']} but for DESI forecast OHD in 4 redshift bins.
• Figure 5: Graphical profile likelihood for the $H_0$ parameter and CC data with $z_{\textrm{min}} = 0$ (full sample). The $68\%$ confidence intervals for three methods are denoted by vertical lines.