Likelihood-ratio asymptotics and boundaries

Consider a correctly specified, fixed-dimensional, dominated model at an identifiable interior point with nonsingular Fisher information. Assume the smoothness, local uniform quadratic likelihood expansion, and consistent full and restricted maximizers needed for regular likelihood asymptotics. If the null is locally a smooth embedded submanifold of codimension $r$, the twice-log-likelihood difference converges to chi-square with $r$ degrees of freedom. These hypotheses exclude a boundary point, unidentified nuisance parameters, singular information, and parameter dimensions growing with sample size.

In information-normalized coordinates, the quadratic likelihood is optimized over a linear tangent space for the null and the full Euclidean space for the alternative. Their difference is a squared Gaussian projection onto the orthogonal complement of the null tangent space. This gives the chi-square law and explains why codimension is the relevant count. The geometry must actually be linear at the point; writing down a nominal number of restrictions does not establish the projection argument.

For a normal mean constrained to be nonnegative, the maximum-likelihood estimator is the positive part of the sample mean. At a null mean of zero, a negative unconstrained estimate cannot improve the fit over the null. The likelihood-ratio statistic is then zero. A positive unconstrained estimate contributes its usual standardized square. Since a centered normal has equal mass on either side of zero, the resulting distribution mixes a point mass and a one-degree chi-square law with equal weights.

Other boundary problems can have different mixture weights, several chi-square components, or entirely different limits. The answer depends on the parameter constraints, information metric, and whether nuisance parameters remain identified. The half-and-half mixture is a result for the specific one-sided scalar geometry here. In a general cone problem, one derives the limiting optimization or invokes a theorem whose assumptions fit that cone; the presence of any boundary does not by itself identify this distribution.

For a positive observed statistic, the upper-tail probability is one half of the corresponding one-degree chi-square tail, equivalently the upper standard-normal tail at its square root. At an observed value of zero, the conventional nonrandomized upper-tail p-value using a greater-than-or-equal event is one: the statistic is always nonnegative. Extending the positive-value formula to zero would ignore the atom and report one half instead. Different randomized conventions must be stated explicitly rather than silently substituted.

For a five-percent test, the correct positive critical value is the square of the ninety-fifth standard-normal percentile. Using the usual ninety-fifth percentile of a one-degree chi-square distribution instead gives a conservative two-and-a-half-percent test in this example. More elaborate boundary models often require specialized asymptotic theory or a carefully validated parametric simulation. A bootstrap is not automatically valid just because the ordinary chi-square rule fails; resampling must reproduce the boundary experiment and its nuisance behavior.