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arxiv: 1007.1727 · v3 · submitted 2010-07-10 · ⚛️ physics.data-an · hep-ex

Asymptotic formulae for likelihood-based tests of new physics

Glen Cowan , Kyle Cranmer , Eilam Gross , Ofer Vitells This is my paper

Pith reviewed 2026-05-11 01:44 UTC · model grok-4.3

classification ⚛️ physics.data-an hep-ex
keywords asymptotic distributionslikelihood ratio testsWilks theoremAsimov datasetsystematic uncertaintiesnew physicsdiscovery significancemedian sensitivity
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0 comments X

The pith

Explicit asymptotic formulae are derived for the distributions of likelihood ratio test statistics in searches for new physics, including systematic uncertainties, along with the Asimov dataset for computing median experimental sensitivity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Physicists need practical ways to assess the significance of potential signals or to set limits on new physics parameters when data are limited and systematic effects are present. This work supplies closed-form expressions for how common test statistics behave in the large-sample limit, based on established statistical results. These expressions replace the need for many Monte Carlo simulations when calculating p-values or expected sensitivities. A central tool is the Asimov dataset, a single representative dataset in which every observed value is set to its expected value, which directly yields the median outcome of an experiment.

Core claim

Using results from Wilks and Wald, explicit formulae are derived for the asymptotic distributions of the likelihood ratio test statistics. These account for systematic uncertainties by profiling over nuisance parameters. The Asimov dataset is motivated and justified as a representative data set that provides the median experimental sensitivity of a search or measurement as well as fluctuations about this expectation.

What carries the argument

The Asimov data set, a single constructed dataset in which observed values equal their expected values under the model hypothesis, which serves as a direct proxy for the median value of the test statistic.

If this is right

  • P-values and significances for discovery can be calculated analytically from the asymptotic distributions without generating pseudo-experiments.
  • Systematic uncertainties enter the test through the profiled likelihood and are automatically included in the asymptotic formulae.
  • The median expected sensitivity or exclusion reach of an experiment is obtained by evaluating the test statistic once on the Asimov dataset.
  • The same machinery supplies the asymptotic behaviour needed to construct confidence intervals on parameters of interest.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method lowers the computational cost of sensitivity studies in analyses where full detector simulations are expensive.
  • Analogous asymptotic shortcuts may prove useful in likelihood-based inference problems outside particle physics that also involve nuisance parameters.
  • The accuracy of the approximations for moderate event counts or near parameter boundaries remains an open question that can be checked case by case.

Load-bearing premise

The profiled likelihood satisfies the regularity conditions of Wilks' and Wald's theorems, including sufficiently large sample size and an interior maximum.

What would settle it

Monte Carlo simulations of a concrete analysis in which the observed distribution of the test statistic under the background-only hypothesis deviates substantially from the predicted asymptotic form, such as a chi-squared distribution with one degree of freedom.

read the original abstract

We describe likelihood-based statistical tests for use in high energy physics for the discovery of new phenomena and for construction of confidence intervals on model parameters. We focus on the properties of the test procedures that allow one to account for systematic uncertainties. Explicit formulae for the asymptotic distributions of test statistics are derived using results of Wilks and Wald. We motivate and justify the use of a representative data set, called the "Asimov data set", which provides a simple method to obtain the median experimental sensitivity of a search or measurement as well as fluctuations about this expectation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript derives explicit asymptotic formulae for the distributions of profile likelihood ratio test statistics for testing new physics signals and constructing confidence intervals in high energy physics, incorporating systematic uncertainties. It uses results from Wilks and Wald to obtain these distributions, including boundary-adjusted forms for non-negative signal strengths, and introduces the Asimov dataset to efficiently calculate the median experimental sensitivity and its fluctuations, supported by comparisons with toy Monte Carlo simulations.

Significance. This work offers practical and widely applicable methods for statistical inference in particle physics searches. The provision of closed-form expressions for the test statistic distributions and the Asimov dataset approach significantly reduces the computational burden for sensitivity studies. The paper's analytic results are validated against numerical simulations, enhancing their reliability for use in experimental analyses at facilities like the LHC. The explicit formulae and reproducible comparisons with toy MC are particular strengths.

minor comments (3)
  1. It would be helpful to specify in the abstract that the formulae account for systematic uncertainties through the inclusion of nuisance parameters in the likelihood.
  2. The presentation of the asymptotic distributions could include a brief note on the conditions under which the chi-squared approximation holds for the profiled likelihood.
  3. The definition of the Asimov dataset is well-motivated, but adding an equation for the expected event counts in the presence of systematics would aid reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful review and for recommending acceptance of the manuscript. The referee's summary and assessment accurately reflect the scope and contributions of the work.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives asymptotic distributions of profile likelihood ratio statistics directly from the external theorems of Wilks (1938) and Wald (1943), with explicit mixture-of-chi-squared forms for boundary cases (μ ≥ 0) obtained by applying those theorems under stated regularity conditions. The Asimov data set is introduced as an explicitly constructed representative dataset whose expected counts yield the median value of the test statistic under the asymptotic limit; this construction is definitional and does not reduce any prediction to a fitted parameter or self-referential input. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation appear in the central derivation chain. The paper supplies both analytic expressions and independent toy-MC validation, confirming the derivations remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper rests on two classical theorems plus the definition of the Asimov dataset; no free parameters are fitted and no new physical entities are postulated.

axioms (2)
  • standard math Wilks' theorem: under regularity conditions the likelihood-ratio statistic is asymptotically chi-squared distributed.
    Invoked in §2 and §3 to obtain the asymptotic null distribution.
  • standard math Wald's theorem: the maximum-likelihood estimator is asymptotically normal with variance given by the inverse Fisher information.
    Used to derive the non-centrality parameter of the test statistic under the alternative.
invented entities (1)
  • Asimov dataset independent evidence
    purpose: Single representative dataset whose likelihood ratio equals the median of the test-statistic distribution under a given hypothesis.
    Defined in §4 as the expected counts under the hypothesis; independent evidence is the numerical agreement with toy-MC medians shown in the paper.

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