Boundary Nulls and Other Failures of Standard Calibration

When the Usual Chi-Square Reference Is Not Safe

A likelihood-ratio statistic can have a familiar chi-square limit only when the model is locally regular near the null. In particular, the null parameter should be an interior point of a locally identifiable model, and the likelihood should admit a suitable quadratic approximation. If a constraint puts the null on the boundary, a parameter disappears under the null, or the model's support changes with the parameter, that approximation can fail or have a different geometry. The statistic may still have a limiting distribution, but the usual degrees of freedom need not describe it.

The earlier material on Wilks' theorem gives the standard benchmark: under suitable regularity conditions, twice the log-likelihood difference between an unrestricted fit and a restricted fit converges in distribution to a chi-square random variable, with degrees of freedom given by the number of independent restrictions. This tutorial focuses on diagnosing cases where those conditions do not hold and seeing concretely what can replace that benchmark.

Regular Wilks calibration: For a nested likelihood-ratio test with $q$ independent restrictions, the usual $\chi^2_q$ limit requires, among other conditions, a correctly specified model that is locally identifiable, a null value in the interior of the parameter space, a nonsingular Fisher information matrix, and a sufficiently smooth likelihood with suitable integrability and common-support properties. When these conditions fail, the ordinary degrees-of-freedom rule is not automatic.

A Scalar Parameter on the Boundary

Consider testing $H_0:\theta=0$ against $\theta\geq 0$ in a normal location model with known variance 1. Under the null, $\theta$ is the endpoint of the allowed parameter space, not an interior value. The unrestricted estimate within the allowed space is $\hat\theta=\max(0,\bar X)$, where $\bar X$ is the sample mean.

Worked Example: A Boundary Null in a Normal Model

Let $X_1,\ldots,X_n$ be independent with $X_i\sim N(\theta,1)$, where $\theta\geq0$. Test $H_0:\theta=0$ against $\theta>0$ using the likelihood-ratio statistic.

Ignoring terms that do not depend on $\theta$, the log-likelihood is $\ell(\theta)=-\frac12\sum_{i=1}^n(X_i-\theta)^2$. Its unconstrained maximizer is $\bar X$; imposing $\theta\geq0$ gives $\hat\theta=\max(0,\bar X)$. The statistic is

T=2\{\ell(\hat\theta)-\ell(0)\} =n\{\max(0,\bar X)\}^2.

Under $H_0$, $Z=\sqrt n\,\bar X\sim N(0,1)$ exactly. Therefore $T=\{\max(0,Z)\}^2$. If $Z\leq0$, which occurs with probability $1/2$, then $T=0$. If $Z>0$, then $T=Z^2$. Thus the null distribution is a mixture: it places probability $1/2$ at zero and probability $1/2$ on a $\chi^2_1$ distribution.

This is a simple instance of a chi-bar-square distribution: a mixture of chi-square distributions that can arise from likelihood problems with boundary constraints. The particular mixture weights depend on the local constraint geometry and other model details; the one-half weights above are not a universal boundary rule.

Derived boundary calibration: In the example above, for any observed $t>0$, $$ P(T\geq t)=\tfrac12 P(\chi^2_1\geq t). $$ The factor $1/2$ accounts for the probability mass at zero, which cannot contribute to an upper-tail event at positive $t$.

For example, using the ordinary 5% upper-tail critical value $3.841$ for $\chi^2_1$ would give a test with null rejection probability $\frac12(0.05)=0.025$ in this model, not 0.05. The 5% critical value for the mixture instead solves $\frac12 P(\chi^2_1\geq c)=0.05$, so $c$ is the 90th percentile of $\chi^2_1$, approximately 2.706. This distinction is consequential: the ordinary critical value is conservative here, but boundary problems do not all have the same correction.

A Parameter That Is Unidentified Under the Null

A different failure occurs when some parameter has no effect on the distribution under the null. A likelihood can then be maximized over values of that parameter even though the null data contain no information about which value is relevant. The resulting statistic may depend on the maximum of a collection of random scores, rather than on one fixed-dimensional chi-square law.

Worked Example: An Unidentified Component in a Mixture

Suppose $X_1,\ldots,X_n$ are independent with density $f(x;\pi,\mu)=(1-\pi)\phi(x;0,1)+\pi\phi(x;\mu,1)$, where $0\leq\pi\leq1$ and $\mu$ is restricted to the two values $-1$ and $1$. Test $H_0:\pi=0$ against $\pi>0$. Under the null, $X_i\sim N(0,1)$, whatever value is assigned to $\mu$. Thus $\mu$ is unidentified under $H_0$.

For a fixed choice of $\mu$, the density relative to the null density is $1+\pi\{r_\mu(x)-1\}$, where $r_\mu(x)=\phi(x;\mu,1)/\phi(x;0,1)=\exp(\mu x-\mu^2/2)$. At $\pi=0$, the score for $\pi$ is therefore $\sum_{i=1}^n\{r_\mu(X_i)-1\}$. Under the null, each summand has mean zero. Its variance is $e-1$ for either choice of $\mu$, since $E[r_\mu(X)^2]=e$ when $X\sim N(0,1)$ and $\mu=\pm1$.

The two scores are not independent. Their covariance for one observation is $E[(r_1(X)-1)(r_{-1}(X)-1)]=e^{-1}-1$, because $E[r_1(X)r_{-1}(X)]=e^{-1}$. Their correlation is $(e^{-1}-1)/(e-1)=-e^{-1}$. If $Z_+$ and $Z_-$ denote the corresponding standardized joint limiting scores, the local quadratic likelihood approximation gives the likelihood-ratio limit

\max\{0,Z_+,Z_-\}^2, \qquad \operatorname{Corr}(Z_+,Z_-)=-e^{-1}.

To see why the maximization appears, for either fixed component the local log-likelihood has the form $hZ-\frac12h^2$ after standardization, with $h\geq0$. Maximizing over $h$ gives $\frac12\{\max(0,Z)\}^2$; doubling the log-likelihood gain gives $\{\max(0,Z)\}^2$. The alternative chooses the larger gain from the two component values. This maximum has a distribution determined by the joint scores, not the ordinary $\chi^2_1$ distribution from counting the single restriction on $\pi$.

The example also shows why simply counting parameters or restrictions is not enough. Under the null, changing $\mu$ changes no distribution at all; under the alternative, it changes the component being fitted. The relevant likelihood behavior involves the competing score directions and their dependence. More complicated unidentified-parameter problems can produce limits involving a supremum of a Gaussian process, and there is no single universal chi-square replacement.

When the Support Depends on the Parameter

Wilks' usual smooth, quadratic likelihood approximation can also break down when the support of the observations changes with the parameter. An endpoint model makes this failure especially transparent. Its likelihood-ratio statistic is still calculable, but its distribution does not follow the usual one-restriction rule.

Worked Example: A Uniform Endpoint Parameter

Let $X_1,\ldots,X_n$ be independent and uniform on $(0,\theta)$, with $\theta>0$. Test $H_0:\theta=\theta_0$ against the unrestricted alternative $\theta>0$, where $\theta_0$ is the true value under the null.

Writing $X_{(n)}=\max_i X_i$, the likelihood is $L(\theta)=\theta^{-n}$ when $X_{(n)}\leq\theta$, and zero otherwise. It decreases as $\theta$ increases over the feasible values, so the maximum-likelihood estimate is $\hat\theta=X_{(n)}$. Under the null, $X_{(n)}\leq\theta_0$ with probability one, and the likelihood-ratio statistic is

T=2\{\ell(\hat\theta)-\ell(\theta_0)\} =2n\log\left(\frac{\theta_0}{X_{(n)}}\right).

For $0<u<1$, $P\{X_{(n)}/\theta_0\leq u\}=u^n$, since all $n$ observations must be at most $u\theta_0$. Consequently, for $t\geq0$,

P(T>t) =P\left\{\frac{X_{(n)}}{\theta_0}<e^{-t/(2n)}\right\} =\left(e^{-t/(2n)}\right)^n =e^{-t/2}.

This is exactly the upper tail of $\chi^2_2$, for every $n$. It is not the $\chi^2_1$ law suggested by counting one restriction. The coincidence with $\chi^2_2$ does not restore Wilks' regularity conditions: the support depends on $\theta$, and the log-likelihood is not a smooth quadratic function in the usual local sense.

A Practical Diagnostic

Before using a standard chi-square reference distribution, check the structure of the null and the likelihood rather than relying only on the number of restrictions. In particular, ask whether the null is interior, whether all parameters are identified under it, and whether the distribution's support changes with the parameter. A failed condition is a warning that the standard calibration needs justification, not proof by itself that every chi-square approximation is wrong; as the uniform example shows, a nonregular model can sometimes yield a chi-square law for a different reason.

Check the parameter-space geometry.
If the null lies on a boundary, determine the local feasible directions. A one-sided scalar constraint can create a point mass at zero; more general constraints can yield other mixture weights.

Check identification under the null.
Ask whether distinct parameter values give the same null distribution. If a nuisance parameter disappears, the likelihood may maximize over a family of score directions.

Check likelihood regularity.
Look for parameter-dependent support, nonsingular information, and a valid local quadratic approximation. If these fail, derive the null distribution for the model or use a method justified for that nonregular setting.

A common mistake is to apply the usual chi-square degrees of freedom simply because the test compares nested models or because a software package reports a likelihood-ratio statistic. The correction is to verify the calibration assumptions first. Boundary mixtures, unidentified-parameter limits, and endpoint-model limits are not interchangeable; each reflects a different likelihood geometry. A simulation or bootstrap can help assess calibration, but it must reproduce the null structure and be justified for the particular problem rather than assumed to repair nonregularity automatically.

Key takeaway: The standard chi-square rule is a consequence of regular local likelihood behavior, not a universal property of likelihood-ratio statistics. Boundary constraints, parameters unidentified under the null, and parameter-dependent support each require their own distributional analysis.

Check Your Understanding

Use the examples and diagnostic questions above to test your understanding.

In the constrained normal model, why does the likelihood-ratio statistic have probability $1/2$ at zero under the null?
For a positive observed statistic in that model, how does the correct mixture upper-tail probability relate to the $\chi^2_1$ upper tail?
In the mixture example, why is the component mean $\mu$ unidentified when $\pi=0$?
Why does counting one restriction not justify using $\chi^2_1$ in the uniform endpoint example?
Name three features of a model or null hypothesis that should prompt a check of standard chi-square calibration.