Boundary Parameters And Nonstandard Chi-Square Limits

Why a Boundary Changes the Likelihood Ratio Limit

Wilks' theorem gives a chi-square approximation for many likelihood ratio tests when the null model is nested in the alternative and the true parameter is an interior point of the allowed parameter space. At a boundary, the likelihood cannot move freely in every local direction: an estimate that would fall outside the allowed space is forced back to its edge. That constraint changes the limiting distribution of the likelihood ratio statistic.

The simplest case is a single parameter \(\theta\) restricted to \(\theta\geq 0\), with the null hypothesis \(H_0:\theta=0\). The null value is allowed, but it is the edge of the parameter space, not an interior point. Under standard regularity conditions, the likelihood ratio statistic then has a limiting distribution with both a point mass at zero and a chi-square component. This is called a chi-bar-square distribution.

Definition: For a one-parameter test of \(H_0:\theta=0\) against \(\theta\geq 0\), the one-sided chi-bar-square distribution is the mixture \(\tfrac12\chi^2_0+\tfrac12\chi^2_1\). Here \(\chi^2_0\) means a point mass at zero, not an ordinary continuous chi-square distribution.

This result concerns an asymptotic null distribution; it need not describe the exact finite-sample distribution. Its use requires a regular likelihood near the null, including positive information for the parameter and a smooth, locally quadratic log-likelihood. The parameter constraint must be a single smooth inequality, and the null value must be on its boundary. If the model is nonregular for other reasons, or the constraint has a different geometry, this particular mixture need not apply.

Deriving the One-Parameter Mixture

Let \(\ell_n(\theta)\) be the log-likelihood based on \(n\) observations, with allowed values \(\theta\geq 0\). The likelihood ratio statistic for \(H_0:\theta=0\) against this constrained alternative is

T_n=2\left\{\sup_{\theta\geq 0}\ell_n(\theta)-\ell_n(0)\right\}.

For a regular model at \(\theta=0\), the local log-likelihood, after scaling the parameter by the square root of the sample size and the Fisher information, has the standard quadratic approximation

\ell_n\left(\frac{h}{\sqrt{nI(0)}}\right)-\ell_n(0) \ \approx\ hZ_n-\frac{h^2}{2}, \qquad h\geq 0,

where \(I(0)\) is the Fisher information per observation at the null and \(Z_n\) converges in distribution to a standard normal random variable \(Z\). The condition \(h\geq 0\) is the local form of the parameter constraint. To maximize the quadratic expression, complete the square:

hZ_n-\frac{h^2}{2} =\frac{Z_n^2}{2}-\frac{(h-Z_n)^2}{2}.

Without the constraint, the maximum occurs at \(h=Z_n\). With \(h\geq 0\), it occurs at \(h=Z_n\) when \(Z_n>0\), and at \(h=0\) when \(Z_n\leq 0\). Thus the constrained maximizer is \(\max(0,Z_n)\). Twice the maximized quadratic increment is \(\{\max(0,Z_n)\}^2\), so

T_n\ \xrightarrow{d}\ \{\max(0,Z)\}^2.

A standard normal variable is negative with probability \(1/2\). On that event, the constrained estimate is at the boundary and the limiting statistic is zero. When \(Z>0\), squaring \(Z\) produces the positive part of a \(\chi^2_1\) distribution. Equivalently, for any \(c>0\),

P(T_n\geq c)\ \longrightarrow\ P(Z\geq\sqrt{c}) =\frac12P(\chi^2_1\geq c).

This is the one-parameter chi-bar-square mixture: half the probability is concentrated at zero, and half follows a \(\chi^2_1\) distribution. If there is a regular nuisance parameter, the same form can apply after accounting for it through the appropriate efficient score and information. It is not correct simply to use the unadjusted score for the parameter of interest when nuisance estimation changes its variance.

Testing Consequences

For a positive observed statistic \(t\), the asymptotic boundary-adjusted p-value is half the ordinary \(\chi^2_1\) upper-tail probability. Since a \(\chi^2_1\) variable is the square of a standard normal variable, this can also be written using the standard normal upper tail. At an observed statistic of zero, the inclusive-tail p-value is one.

p_{\text{boundary}}(t) =\frac12P(\chi^2_1\geq t) =1-\Phi(\sqrt{t}),\qquad t>0.

For a test of asymptotic size \(\alpha<1/2\), the rejection rule is to reject for \(T_n>c_\alpha\), where \(c_\alpha\) is the \((1-2\alpha)\)-quantile of \(\chi^2_1\). For instance, at \(\alpha=0.05\), the boundary critical value is about 2.7055, the 90th percentile of \(\chi^2_1\). The corresponding standard normal cutoff is about 1.6449.

Using the ordinary 5% \(\chi^2_1\) critical value of about 3.841 instead gives an asymptotic rejection probability of only 2.5% under this boundary null. It is therefore conservative at this boundary. The adjustment matters for both reported p-values and test size, even when it does not change a particular decision.

Worked Example: A Constrained Normal-Mean Test

Suppose \(X_1,\ldots,X_{25}\) are independent observations from \(N(\theta,25)\), where the variance \(25\) is known and the parameter space is \(\theta\geq 0\). Test \(H_0:\theta=0\) against \(\theta\geq 0\). The observed sample mean is \(\bar X=1.8\). Find the likelihood ratio statistic, its boundary-adjusted asymptotic p-value, and the 5% decision. This method uses the constrained maximum likelihood estimate and the chi-bar-square calibration; it avoids treating the boundary null as an interior point.

The log-likelihood, apart from a constant, is \(-\tfrac{1}{2(25)}\sum_{i=1}^{25}(X_i-\theta)^2\). Without the constraint, the maximum likelihood estimate is \(\bar X=1.8\). Since this is allowed, the constrained estimate is also \(\hat\theta=1.8\).

The likelihood ratio statistic is the reduction in the residual sum of squares, divided by the known variance:

T=\frac{n(\bar X-0)^2}{25} =\frac{25(1.8)^2}{25} =3.24.

The normal statistic is \(Z=\sqrt{T}=1.8\). Therefore the mixture p-value is \(1-\Phi(1.8)\), which is approximately \(0.0359\). The 5% boundary critical value is 2.7055, and \(3.24>2.7055\), so reject \(H_0\) at the 5% level.

As a calibration check, the ordinary \(\chi^2_1\) tail probability at 3.24 would be about \(0.0719\), twice the boundary-adjusted value. Applying the ordinary 5% critical value 3.841 would fail to reject. The mismatch arises because the ordinary calibration omits the boundary's point mass at zero.

Worked Example: A Binomial Parameter at Its Lower Bound

A hypothetical quality-control process records \(X=30\) defective components among \(n=50\) independent components. Model \(X\sim\operatorname{Binomial}(50,p)\), with the restricted alternative \(p\geq 0.5\), and test \(H_0:p=0.5\). Find the constrained estimate and likelihood ratio statistic, then give the asymptotic boundary p-value. This calculation uses the binomial likelihood directly and avoids assuming that the unconstrained estimate is always admissible.

The unconstrained maximum likelihood estimate is \(\hat p_{\mathrm{unconstrained}}=X/n=30/50=0.6\). Since \(0.6\) satisfies the restriction \(p\geq 0.5\), it is also the constrained estimate. The binomial coefficient cancels in the likelihood ratio, giving

T=2\left[ 30\ln\left(\frac{0.6}{0.5}\right) +20\ln\left(\frac{0.4}{0.5}\right) \right].

The two logarithm terms are \(30\ln(1.2)=5.46965\) and \(20\ln(0.8)=-4.46287\). Their sum is \(1.00678\), so \(T=2.01355\). The statistic is positive, and the boundary-adjusted asymptotic p-value is

1-\Phi(\sqrt{2.01355}) =1-\Phi(1.4190) \approx 0.0780.

At the 5% level, the p-value exceeds \(0.05\), so the test does not reject. The ordinary \(\chi^2_1\) tail at this statistic would be about \(0.1559\), twice the mixture p-value. Both calculations lead to the same decision here, but only the mixture calibration matches the stated boundary setup asymptotically.

Worked Example: When the Constrained Estimate Is at the Boundary

In another hypothetical inspection, \(X=20\) of \(n=80\) items are defective. Let \(X\sim\operatorname{Binomial}(80,p)\), restrict \(p\geq 0.3\), and test \(H_0:p=0.3\). Find the constrained maximum likelihood estimate, the likelihood ratio statistic, and its inclusive-tail boundary p-value. This example uses the same likelihood method but highlights the point mass at zero.

The unconstrained estimate is \(20/80=0.25\), which is not allowed under \(p\geq 0.3\). The binomial log-likelihood increases up to the unconstrained estimate and decreases after it, so its maximum over the restricted parameter space occurs at the nearest allowed value, \(\hat p=0.3\).

The null value is also \(p=0.3\). Consequently, the maximized likelihood under the alternative equals the likelihood under the null, and

T=2\{\ell(0.3)-\ell(0.3)\}=0.

Because the observed statistic is zero, its inclusive-tail p-value is \(P(T\geq 0)=1\). The test does not reject at 5%. The common mistake is to substitute the inadmissible estimate \(0.25\) into the alternative likelihood; doing so would compare the null with a parameter value outside the stated model and produce an invalid statistic.

Common Calibration Traps

Boundary likelihood ratio tests require attention to both the parameter space and the direction of the alternative. The following checks prevent frequent errors:

Check whether the null is interior. Wilks' ordinary chi-square result requires regular interior conditions. A null value at the edge of the allowed space can instead produce a mixture limit.
Constrain the estimate before calculating the statistic. For a one-sided parameter space, an unconstrained estimate on the wrong side is replaced by the boundary estimate. It is not a valid alternative estimate.
Use the mixture tail, not the full chi-square tail. For \(t>0\), the one-parameter boundary p-value is half the \(\chi^2_1\) upper tail.
Do not overlook the atom at zero. A zero statistic is a possible limiting outcome with probability \(1/2\); its inclusive-tail p-value is one.
Do not generalize the weights without checking the constraint. Multiple boundary parameters or more complicated constraints can lead to different chi-bar-square weights. The one-parameter half-and-half mixture is not a universal boundary rule.

Recognizing the boundary changes the reference distribution, not the definition of the likelihood ratio statistic itself. Correct calibration is essential when a model parameter is constrained to be nonnegative, when a null value is an endpoint, or when a one-sided alternative makes the null a boundary point. The distinction is aligned with the SOA VEE Mathematical Statistics syllabus.

Check Your Understanding

For each problem, use the one-parameter boundary calibration when its stated assumptions apply. Answers are shown so that you can check both the statistic and the conclusion.

Let \(X_1,\ldots,X_{36}\) be independent \(N(\theta,9)\) observations, with known variance and \(\theta\geq 0\). Test \(H_0:\theta=0\) when \(\bar X=0.5\). Find \(T\) and the asymptotic boundary p-value. (Answer: \(T=1.000\), \(p=1-\Phi(1)=0.1587\).)
Let \(X_1,\ldots,X_{25}\) be independent \(N(\theta,16)\) observations, with \(\theta\geq 0\), and suppose \(\bar X=-0.4\). Find the constrained estimate and the likelihood ratio statistic. (Answer: \(\hat\theta=0\), \(T=0\).)
For a one-parameter boundary likelihood ratio test, find the asymptotic 5% critical value and its equivalent positive standard-normal cutoff. (Answer: \(c_{0.05}=\chi^2_{1,0.90}\approx2.7055\), equivalent cutoff \(z\approx1.6449\).)
Suppose \(X_1,\ldots,X_{49}\) are independent \(N(\theta,49)\), with known variance and \(\theta\geq0\), and \(\bar X=2\). Test \(H_0:\theta=0\) at 5%. (Answer: \(T=4.000\), \(p=1-\Phi(2)=0.0228\); reject using the boundary critical value 2.7055.)
Let \(X\sim\operatorname{Binomial}(50,p)\), with the restricted alternative \(p\geq0.5\), and observe \(X=25\). Find the constrained estimate and likelihood ratio statistic. (Answer: \(\hat p=0.5\), \(T=0\), and the inclusive-tail p-value is 1.)

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