Boundary and Nonexistence of Maximum-Likelihood Estimates

When the Likelihood Has No Finite Maximizer

Likelihood-based inference usually begins by maximizing \(L(\theta)\), or equivalently its log \(\ell(\theta)\). In a generalized linear model, numerical algorithms can return very large coefficients or fail to settle down when the likelihood keeps improving as a coefficient moves toward infinity. A different issue arises when distinct coefficient vectors give exactly the same fitted values: the likelihood can have a finite maximum but cannot select a unique estimate. These are different failures and have different implications.

The distinction matters because standard large-sample procedures typically use an estimate near a finite, identifiable parameter and a nonsingular information matrix. A boundary limit or a flat direction can break those conditions. A numerical stopping point, by itself, does not establish that a maximum exists.

Three Different Likelihood Situations

Definition: A finite interior maximum is attained at a finite parameter value lying in the interior of the parameter space. A boundary maximum is attained at a point on the parameter-space boundary. A supremum not attained at a finite parameter occurs when likelihood values can approach their least upper bound along a sequence of finite parameter values, but no finite value attains it. A parameter is nonidentifiable in a model if different parameter values induce the same distribution; in a GLM, a rank-deficient design can make different coefficient vectors give the same linear predictors and likelihood.

The parameterization matters. For a Bernoulli observation with success probability \(p\), the likelihood is defined at \(p=0\). If the observation is a failure, \(p=0\) is a boundary maximum. Under the logit parameterization \(p=\exp(\beta)/(1+\exp(\beta))\), however, \(p=0\) corresponds to \(\beta=-\infty\), not to a finite coefficient. Thus a model can have a boundary maximum in the probability parameterization but no finite maximum in its logit parameterization.

For a canonical-link GLM with independent responses, fixed positive dispersion, and linear predictors \(\eta_i=x_i^T\beta\), the part of the log-likelihood that depends on \(\beta\) has the form

\ell(\beta)=\frac{1}{a(\phi)}\sum_{i=1}^n \{y_i\eta_i-b(\eta_i)\}+\text{constant},

where \(a(\phi)>0\). The parameter space for \(\beta\) may be all of \(\mathbb{R}^p\), but that does not guarantee a finite maximizer. The likelihood can keep increasing along a direction in \(\mathbb{R}^p\), or it can be constant along a direction because the design does not distinguish some coefficients.

Separation in Logistic Regression

For independent Bernoulli responses \(y_i\in\{0,1\}\), logistic regression has log-likelihood

\ell(\beta)=\sum_{i=1}^n \left\{y_i x_i^T\beta-\log\left(1+\exp(x_i^T\beta)\right)\right\}.

A direction \(d\) separates the observations if \((2y_i-1)x_i^T d\geq 0\) for every \(i\). If at least one of these inequalities is strict, moving the coefficient vector along \(d\) improves the likelihood at every finite point. This includes complete separation, in which all inequalities are strict, and quasi-complete separation, in which some are equalities.

Theorem: In the logistic model above, if there is a vector \(d\) such that \((2y_i-1)x_i^T d\geq 0\) for all \(i\), with strict inequality for at least one \(i\), then \(\ell(\beta)\) has no finite maximizer.

Proof. Fix any finite \(\beta\) and let \(p_i(t)=\exp(x_i^T(\beta+td))/(1+\exp(x_i^T(\beta+td)))\). Differentiating along the line gives

\frac{d}{dt}\ell(\beta+td) =\sum_{i=1}^n \{y_i-p_i(t)\}x_i^T d.

If \(y_i=1\), the separation condition says \(x_i^T d\geq 0\), so that term is \((1-p_i(t))x_i^T d\geq 0\). If \(y_i=0\), it says \(x_i^T d\leq 0\), so that term is \(-p_i(t)x_i^T d\geq 0\). At least one term is strictly positive for finite \(t\), because \(0<p_i(t)<1\) and at least one margin is strict. Therefore the derivative is positive for every finite \(t\). In particular, from any finite \(\beta\), a sufficiently small positive step along \(d\) increases the log-likelihood, so no finite \(\beta\) can maximize it. \(\square\)

Worked Example: Complete Separation With Two Observations

Consider two Bernoulli observations with an intercept and one predictor: \((x_1,y_1)=(-1,0)\) and \((x_2,y_2)=(1,1)\). Write \(\beta=(\beta_0,\beta_1)^T\). The direction \(d=(0,1)^T\) gives \(x_1^Td=-1\) and \(x_2^Td=1\). For the first observation, \((2y_1-1)x_1^Td=(-1)(-1)=1\); for the second, \((2y_2-1)x_2^Td=(1)(1)=1\). Both margins are strict, so the theorem rules out a finite maximizer.

The log-likelihood along \(\beta=(0,t)^T\) is

\ell(0,t)=-\log(1+\exp(-t))-\log(1+\exp(-t)) =-2\log(1+\exp(-t)).

At \(t=0\), this is \(-2\log 2\). As \(t\to\infty\), \(\exp(-t)\to 0\), so \(\ell(0,t)\to 0\). Each Bernoulli log-likelihood contribution is at most zero, so zero is the supremum; it is not attained at any finite \(t\). The fitted probabilities approach zero for the failure and one for the success, while the slope diverges.

A quasi-separating direction can also prevent a finite maximum. Observations with \(x_i^Td=0\) keep their linear predictors unchanged along the direction; the strict margins still make the total log-likelihood increase. The limiting likelihood need not be the perfect-fit limit of complete separation.

Boundary Limits in a Poisson Model

Worked Example: All-Zero Counts Under a Log Link

Suppose \(Y_1,\ldots,Y_n\) are independent Poisson counts with means \(\mu_i=E_i\exp(\beta)\), where each known exposure \(E_i\) is positive. If every observed count is zero, then, up to a constant that does not depend on \(\beta\),

\ell(\beta)=-\sum_{i=1}^n E_i\exp(\beta).

For every finite \(\beta\), the derivative is \(-\exp(\beta)\sum_i E_i<0\). Hence decreasing \(\beta\) increases the log-likelihood. Its limit as \(\beta\to-\infty\) is zero, and \(\ell(\beta)<0\) for every finite \(\beta\). The supremum is zero but is not attained at a finite log-mean coefficient.

If the model is instead expressed in terms of the common rate multiplier \(\lambda=\exp(\beta)\) and extended to allow \(\lambda=0\), the likelihood is maximized at the boundary \(\lambda=0\). This is the same distinction as in the Bernoulli example: a boundary estimate in a closed mean-parameter space can correspond to an infinite coefficient in a link-scale parameterization.

An infinite estimate is not a large but ordinary finite estimate. A reported value such as \(-25\) may be a numerical stopping point on a path where the likelihood is still increasing. Estimates, standard errors, and tests computed at that point can depend strongly on numerical tolerances.

Nonidentifiability and Nonunique Estimates

Worked Example: Duplicate Design Columns

Take two Bernoulli observations, both with design row \(x_i^T=(1,1)\), and outcomes \(y_1=0\), \(y_2=1\). Both linear predictors equal \(s=\beta_1+\beta_2\). Thus

\ell(\beta_1,\beta_2) =s-2\log(1+\exp(s)).

Differentiating with respect to \(s\) gives \(1-2\exp(s)/(1+\exp(s))\). Setting this to zero yields \(1+\exp(s)=2\exp(s)\), so \(s=0\). The second derivative is \(-2\exp(s)/(1+\exp(s))^2<0\), confirming a maximum in \(s\). The maximum log-likelihood is \(-2\log 2\).

Every coefficient pair satisfying \(\beta_1+\beta_2=0\) attains that same maximum. For example, \((0,0)\), \((1,-1)\), and \((-3,3)\) all give identical fitted probabilities of \(1/2\). The coefficients are not uniquely identified, even though the fitted probabilities and the maximum likelihood value are.

More generally, if a nonzero vector \(v\) satisfies \(Xv=0\), then \(X(\beta+tv)=X\beta\) for every real \(t\). In a model whose likelihood depends on \(\beta\) only through \(X\beta\), it follows directly that \(\ell(\beta+tv)=\ell(\beta)\). This proves that a unique coefficient vector cannot be recovered from the likelihood. It does not, by itself, imply that the fitted means are nonunique or that the likelihood has no maximum.

For a concave GLM log-likelihood, a finite stationary point is a global maximum. But that fact does not ensure such a point exists, nor that it is unique. Separation can rule out finite attainment; rank deficiency can make a maximum nonunique. A numerical algorithm may encounter both issues, and its convergence status should be interpreted alongside the model and design.

Consequences for Standard Inference

The usual Wald, score, and likelihood-ratio reference distributions rely on regularity conditions. In a common regular setting, the true parameter is identifiable and interior, the information matrix is nonsingular, the likelihood is sufficiently smooth, and the maximum-likelihood estimator is finite and asymptotically normal. If the estimate is driven toward infinity or the information matrix is singular, the ordinary derivations of standard errors and chi-square reference distributions may not apply.

Wald inference needs a finite estimate and a usable estimate of its covariance. Diverging coefficients or singular information can make Wald statistics unstable, undefined, or misleading.
Likelihood-ratio inference requires meaningful maximized likelihoods under both models. A supremum can sometimes be used, but the standard chi-square limit is not automatic when parameters are on boundaries or are nonidentifiable.
Score inference is computed under the null rather than at the unrestricted estimate. It can remain available when the null model is regular, but the usual score-test result still requires its own identifiability and information conditions; it is not a universal remedy for separation.

A boundary parameter does not always have the ordinary chi-square likelihood-ratio limit. In some regular one-sided boundary problems, a likelihood-ratio statistic has a mixture limit, such as a point mass at zero combined with a chi-square distribution. That conclusion requires the particular model and boundary conditions; it should not be inferred merely from noticing that an estimate is near a boundary. In nonidentifiable or separated models, the limiting behavior can require a different analysis altogether.

Common Pitfall

Common mistake: “The fitting routine returned a large coefficient and a small standard error, so the ordinary Wald test is valid.” A large coefficient may be a finite numerical approximation to a likelihood supremum at infinity, and a local curvature calculation at that stopping point does not establish existence, identifiability, or asymptotic normality. Check for separation, boundary limits, and rank deficiency before interpreting the routine’s output. A finite maximum on the boundary is also not the same as a finite interior maximum, so interior-based reference distributions need separate justification.

When a warning or extreme estimate occurs, inspect the response patterns against the design, check the rank of the design matrix, and distinguish the parameterization in which the model is being fit from the underlying mean parameter. Then decide which inferential conditions actually hold rather than relying on a default reference distribution.

Check Your Understanding

Use the likelihood and model structure, not just an optimizer’s output, to answer these questions.

In logistic regression, what directional condition identifies separation, and why does a strict margin prevent a finite maximizer?
For all-zero Poisson counts under a log link, what happens to the log-likelihood as the intercept tends to negative infinity?
How can a model have a finite maximum likelihood but no unique coefficient estimate?
Why can a boundary maximum in a probability parameter correspond to no finite maximum in a logit coefficient?
Name two regularity conditions that may fail under separation or nonidentifiability, and explain one consequence for standard inference.