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Universal higher-order Bartlett correction

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  • Noma, Hisashi

Abstract

We develop a general higher-order expansion for the likelihood ratio statistic that extends the classical Bartlett–Lawley theory to arbitrary order. Building on a Hermite–cumulant representation of profile and adjusted profile likelihood derivatives, we derive a universal recursive formulation for Bartlett-type correction coefficients, expressed as polynomials in standardized cumulants. This framework unifies first-order Bartlett corrections, Cox–Reid adjusted likelihoods, and higher-order asymptotic refinements within a single algebraic structure. We show that the m-th-order corrected statistic achieves O(n−(m+1)) mean accuracy under standard regularity conditions, and that the same universal coefficients apply to Cox–Reid adjusted likelihoods. Observed-information plug-in versions retain the stated order of accuracy. The results provide a systematic route to higher-order chi-square calibration and confidence intervals for a wide class of likelihood-based procedures.

Suggested Citation

  • Noma, Hisashi, 2026. "Universal higher-order Bartlett correction," Statistics & Probability Letters, Elsevier, vol. 237(C).
    • Handle: RePEc:eee:stapro:v:237:y:2026:i:c:s0167715226001823
    DOI: 10.1016/j.spl.2026.110818
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