# Wilks' theorem

**Wilks' theorem** is a result in mathematical statistics stating that, for two nested parametric models and under suitable regularity conditions, the likelihood ratio test statistic follows an asymptotic chi-square (χ²) distribution as the sample size grows, with degrees of freedom equal to the difference in the number of free parameters between the two models<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup>. It was established by [Samuel S. Wilks](https://www.edgepedia.org/samuel-s-wilks), an American mathematical statistician at Princeton University, in his 1938 paper *The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses*<sup>[2](https://cir.nii.ac.jp/crid/1362262943742806656)</sup>. The theorem supplies the reference distribution that makes the likelihood ratio test (LRT) usable in practice: an observed statistic can be judged against a χ² distribution rather than against a distribution that must be worked out for each problem<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup>.

| Key fact | Detail |
|---|---|
| Statement | The LRT statistic λ_N is asymptotically χ² under the restricted (null) model, with degrees of freedom equal to the difference in the number of free parameters between the nested models<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup> |
| Statistic | 𝟌𝑛 = 2( sup over the full parameter space of the log-likelihood minus sup over the null subspace )<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup> |
| Origin | Wilks (1938), *The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses*<sup>[2](https://cir.nii.ac.jp/crid/1362262943742806656)</sup> |
| Regularity conditions | True parameter value interior to the parameter space; expected Fisher information strictly positive definite; smooth likelihood, finite Fisher information, and correct model specification<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup><sup> • </sup><sup>[3](https://cttir.github.io/tutorials/tutorials/statistical-foundations/likelihood-ratio-tests.html)</sup> |
| Main failures | Boundary problems, indeterminate parameter problems (non-identifiable parameters, singular information matrices), and change-point problems<sup>[4](https://www.research.unipd.it/handle/11577/3441276)</sup> |
| Scope limit | Applies to nested hypotheses; for non-nested models the LRT is not applicable and information criteria or Vuong's test are used instead<sup>[5](https://www.princeton.edu/~erp/ERParchives/archivepdfs/M140.pdf)</sup><sup> • </sup><sup>[3](https://cttir.github.io/tutorials/tutorials/statistical-foundations/likelihood-ratio-tests.html)</sup> |
| Small-sample remedy | Bartlett correction, a rescaling aimed at making the statistic's mean exactly equal to the mean of the asymptotic χ² distribution<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC10997343/)</sup> |

## The statistic and the theorem

For data \( X_1, \dots, X_N \) with log-likelihood \( l_N(\boldsymbol{\theta}) \), the likelihood ratio test statistic is defined as

\[ \lambda_N = 2\left( \sup_{\boldsymbol{\theta} \in \Theta} l_N(\boldsymbol{\theta}) - \sup_{\boldsymbol{\theta} \in \Theta_0} l_N(\boldsymbol{\theta}) \right), \]

where \( \Theta \) is the full parameter space and \( \Theta_0 \) the subspace specified by the null hypothesis<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup>. The null model is nested within the alternative: it is the alternative with one or more parameters constrained to a subset of the parameter space<sup>[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC8293863/)</sup>.

Under suitable regularity conditions, \( \lambda_N \) is asymptotically χ² distributed with degrees of freedom equal to the difference in the number of free parameters between the two models<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup>. In practice the test is conducted by comparing the statistic with this asymptotic distribution under the restricted model<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup>. A related statement, described in one account as the likelihood ratio test of Wilks (1938), is that minus twice the log ratio of two maximized likelihoods follows a χ² distribution with degrees of freedom given by the difference in the number of maximized parameters<sup>[8](https://pmc.ncbi.nlm.nih.gov/articles/PMC12624496/)</sup>.

Wilks' original 1938 paper treated the multivariate normal case: for samples from p-variate normal distributions, the statistic \( W_n \) is asymptotically χ² with \( f = p(p+3)/2 \) degrees of freedom<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC10997343/)</sup>. Wilks' and Wald's (1943) theorems together provide the asymptotic distribution of a function of the likelihood ratio under both null and alternative hypotheses, with estimated parameters taken into account, which extends the test to composite models with unknown parameters<sup>[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC8293863/)</sup>.

## Regularity conditions

The theorem requires several conditions; a standard reference is Theorem 12.4.2 of Lehmann and Romano (2006)<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup>. Two recur in applications:

1. The true parameter value \( \boldsymbol{\theta}^{\ast} \) must be an interior point of \( \Theta \). This fails when the null model lies on the boundary of the parameter space, for example when a variance parameter \( \sigma^2 \) is restricted to \( [0, \infty) \) and the null sets it to zero<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup>.
2. The expected [Fisher information](https://www.edgepedia.org/fisher-information) matrix at \( \boldsymbol{\theta}^{\ast} \), \( I(\boldsymbol{\theta}^{\ast}) = E_{\boldsymbol{\theta}^{\ast}}[\nabla l_N(\boldsymbol{\theta}^{\ast}) \nabla l_N(\boldsymbol{\theta}^{\ast})^{\top}]/N \), must be strictly positive definite. This fails when parameters are non-identifiable or the information matrix is singular<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup>.

A tutorial summary lists the conditions as: the null model \( M_0 \) lying in the interior of the parameter space of the alternative \( M_1 \); regularity in the sense of a smooth likelihood and finite Fisher information; and correct model specification for \( M_1 \)<sup>[3](https://cttir.github.io/tutorials/tutorials/statistical-foundations/likelihood-ratio-tests.html)</sup>.

## When the theorem fails

A 2021 review by Alessandra R. Brazzale, a statistician at the University of Padova, and Valentina Mameli identifies three main classes of problems in which the regularity conditions underlying likelihood-based inference fail: boundary problems, indeterminate parameter problems, which include non-identifiable parameters and singular information matrices, and change-point problems<sup>[4](https://www.research.unipd.it/handle/11577/3441276)</sup>.

**Boundary parameters.** When a parameter of interest or a nuisance parameter lies on the boundary of the parameter space, the asymptotic distribution of the LRT differs from the Wilks χ². This situation arises in variance components testing in many natural science applications and in one-sided tests, and the correct asymptotic distribution must be used whenever it occurs, especially with nuisance parameters also on the boundary<sup>[9](https://onlinelibrary.wiley.com/doi/10.5402/2012/872956)</sup>. In models with latent variables, the regularity conditions can be violated in both ways: a null model on the boundary of the parameter space, or a singular Fisher information matrix under the two-factor parameterization. In one documented example the information matrix is singular when a one-factor model is compared with a two-factor model, but strictly positive definite when the same one-factor model is compared with the saturated model<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup>.

**Non-nested hypotheses.** The standard theory is not applicable when the null and alternative hypotheses are non-nested, meaning neither model is a special case of the other<sup>[5](https://www.princeton.edu/~erp/ERParchives/archivepdfs/M140.pdf)</sup>. In such comparisons the LRT is not applicable, and information criteria such as AIC and BIC or Vuong's test are used instead<sup>[3](https://cttir.github.io/tutorials/tutorials/statistical-foundations/likelihood-ratio-tests.html)</sup>. This limitation matters in searches for new physical phenomena in astrophysics, where competing models are typically non-nested and the usual LRT approach based on asymptotic properties cannot be applied directly<sup>[10](https://arxiv.org/abs/1509.01010)</sup>.

## Extensions and related results

**Bartlett correction.** When p stays fixed while n increases, Bartlett (1937) introduced the first adjustment to the statistic \( W_n \), a rescaling intended to bring its expectation into exact agreement with that of the limiting χ² distribution, namely f.<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC10997343/)</sup> Bartlett-type corrections remain an active tool; a recent application is a correction for likelihood ratio tests of the equality of two Gaussian graphical models in the multivariate normal two-sample problem<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC10997343/)</sup>. Corrections of this kind are also used for small-sample likelihood ratio tests in logistic regression<sup>[11](https://www.tandfonline.com/doi/abs/10.1080/03610926.2017.1373815)</sup>.

**Comparison with Wald and score tests.** The likelihood ratio test belongs to a triad with Rao's score test (1948) and the [Wald test](https://www.edgepedia.org/wald-test) (1941, 1943). A comparison of their sensitivity to changes in nuisance parameters found that, with an error of magnitude \( O(n^{-1}) \), the null distributions and the local alternative distributions of the three tests are equally sensitive to nuisance parameters<sup>[12](https://www.sciencedirect.com/science/article/abs/pii/S0378375800003451)</sup>.

**Empirical likelihood.** The theorem's conclusion extends beyond fully parametric models. [Empirical likelihood](https://www.edgepedia.org/empirical-likelihood), a non-parametric likelihood built by probability profiling of the data, reproduces the Wilks phenomenon of χ² limits for log-ratio statistics for independent data, often without explicit assumptions about the data-generating mechanism, and this carries over to special time series problems<sup>[13](https://www.sciencedirect.com/science/article/pii/S0378375813002528)</sup>.

**Grouped data.** A Wilks-type theorem has also been developed for grouped data, where n is large but only a small amount of observations concentrated in short time intervals are relevant; the resulting procedure uses multiple likelihood ratio statistics over a time window and avoids an arbitrary choice of the origin of time<sup>[14](https://arxiv.org/abs/1802.01715)</sup>.

## Use in practice

The theorem's practical role is to give the likelihood ratio test a known reference distribution for large samples, so that a p-value can be read from the χ² distribution with the appropriate degrees of freedom<sup>[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)</sup>. The result is asymptotic: outside special cases the statistic does not exactly follow a χ² distribution at finite sample sizes, which is why small-sample corrections such as Bartlett's exist<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC10997343/)</sup>. When its conditions hold, the LRT, Wald, and score tests are asymptotically equivalent in the sense above; when they fail, at boundaries or with singular information, the Wilks χ² reference is replaced by the correct nonstandard asymptotic distribution<sup>[9](https://onlinelibrary.wiley.com/doi/10.5402/2012/872956)</sup><sup> • </sup><sup>[4](https://www.research.unipd.it/handle/11577/3441276)</sup>.

## References

1. [A Note on Likelihood Ratio Tests for Models with Latent Variables, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC7826319/)
2. [S. S. Wilks, The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses, CiNii Research](https://cir.nii.ac.jp/crid/1362262943742806656)
3. [Likelihood Ratio Tests tutorial, CTTI](https://cttir.github.io/tutorials/tutorials/statistical-foundations/likelihood-ratio-tests.html)
4. [A. R. Brazzale and V. Mameli (2021), Likelihood Asymptotics in Nonregular Settings: A Review and Annotated Bibliography with Emphasis on the Likelihood Ratio, University of Padova research repository](https://www.research.unipd.it/handle/11577/3441276)
5. [Testing Nonnested Hypotheses, Econometric Reviews](https://www.princeton.edu/~erp/ERParchives/archivepdfs/M140.pdf)
6. [A Bartlett-type correction for likelihood ratio tests with application to testing equality of Gaussian graphical models, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC10997343/)
7. [Errors in Statistical Inference Under Model Misspecification: Evidence, Hypothesis Testing, and AIC, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC8293863/)
8. [There are no alternative hypotheses in tests of null hypotheses, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC12624496/)
9. [L. Kopylev (2012), Constrained Parameters in Applications: Review of Issues and Approaches, International Scholarly Research Notices](https://onlinelibrary.wiley.com/doi/10.5402/2012/872956)
10. [A method for comparing non-nested models with application to astrophysical searches for new physics, arXiv](https://arxiv.org/abs/1509.01010)
11. [Corrected likelihood-ratio tests in logistic regression using small-sample data, Communications in Statistics – Theory and Methods](https://www.tandfonline.com/doi/abs/10.1080/03610926.2017.1373815)
12. [Sensitivity of Rao's score test, the Wald test and the likelihood ratio test to nuisance parameters, Computational Statistics & Data Analysis](https://www.sciencedirect.com/science/article/abs/pii/S0378375800003451)
13. [A review of empirical likelihood methods for time series, Computational Statistics & Data Analysis](https://www.sciencedirect.com/science/article/pii/S0378375813002528)
14. [A Wilks' theorem for grouped data, arXiv](https://arxiv.org/abs/1802.01715)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Statistical inference, estimation, sampling, and testing › Foundations of statistical inference › Asymptotic theory of statistics › Asymptotics of hypothesis tests*

*Initially written Sep 17, 2026 · Reviewed: Sep 19, 2026; Oct 15, 2026 · Edited: Sep 19, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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