Likelihood-ratio tests and pivots

For a null subset $\Theta_0\subseteq\Theta$, define $\Lambda$ as the supremum likelihood under the null divided by the supremum over the full model. This convention makes the ratio at most one when both suprema are positive and finite. Large evidence against the null corresponds to a small ratio or a large statistic $D=-2\log\Lambda$. Nuisance parameters must be optimized within each parameter set rather than silently fixed at convenient values. Numerical failure to find the maxima is a computational concern distinct from statistical calibration.

For normal observations with known variance and an unknown mean, the unrestricted maximizing mean is the sample average. Under the point null the mean is fixed. Expanding the sums of squared residuals cancels the within-sample variation and leaves the squared distance between sample mean and null mean, scaled by its variance. In this special model the likelihood-ratio statistic is exactly the square of a standard-normal variable under the null. The chi-square distribution here is exact, not an application of an asymptotic theorem.

A pivot is a function of data and parameter whose distribution does not depend on unknown parameters in the model. It need not be a statistic, because it may include the unknown parameter. The standardized normal sample mean with known variance is a pivot; replacing the standard deviation by the sample standard deviation yields the Student pivot under iid normal sampling, with $n-1$ degrees of freedom for $n\ge2$. Their parameter-free reference laws support exact test and interval calibration.

Exact pivotal behavior depends on the distributional assumptions. The Student expression need not have a Student distribution under arbitrary heavy-tailed or dependent sampling. An asymptotically pivotal statistic instead approaches a parameter-free reference distribution after taking a specified limit. This can be useful, but its calibration error at a chosen sample size remains a question. A simulation study at several parameter values can investigate that error for those values; it does not prove uniform validity across an unrestricted model.

One sufficient route to Wilks's conclusion is an iid correctly specified finite-dimensional model with an interior identifiable true parameter, nonsingular finite information, a locally uniform quadratic likelihood expansion with a normal limiting score, and consistent constrained and unconstrained maximizers governed by that local expansion. Assume the null is locally a smooth embedded submanifold through the truth, with full-rank constraint derivative and fixed codimension $r$. The quadratic approximation turns the fitted-likelihood difference into the squared information-orthogonal normal component of the limiting score, yielding a chi-square limit with $r$ degrees of freedom. This describes the proof mechanism, not a check-free recipe.

The conclusion is convergence in distribution as sample size grows at a fixed regular null parameter. It is not an exact finite-sample law. A boundary null produces a different local geometry; unidentified mixture components, singular information, or growing parameter dimension also escape the stated assumptions. Even a regular Bernoulli model can have poor small-sample chi-square calibration because its counts are discrete. When feasible, enumerate an exact null distribution or use a separately justified calibration method instead of treating the asymptotic label as a finite-sample guarantee.