Nuisance Parameters Absent Under The Null

Why an Unidentified Parameter Changes the Test

A likelihood-ratio test often compares a restricted model with a larger model. When the null hypothesis fixes a parameter at the boundary of its allowable range, the usual chi-square approximation may fail. A further complication arises when another parameter has no meaning under the null. In that case, the test is not simply moving in one fixed direction away from the null: it is searching across possible directions.

A mixture model makes this issue concrete. Suppose hypothetical claim severities \(X_1,\ldots,X_n\) are independent and have density \(f(x;\alpha,\eta,\beta)=(1-\alpha)f_0(x;\beta)+\alpha f_1(x;\eta)\). Here \(f_0\) is the baseline severity density, \(\beta\) is its parameter, \(\alpha\in[0,1]\) is the mixture weight, and \(\eta\) controls the added component. The null hypothesis is \(H_0:\alpha=0\). Under this null, the density is \(f_0(x;\beta)\), regardless of \(\eta\). Thus the data under the null contain no information about \(\eta\): it is unidentified under the null.

This is different from a test in which a single, identifiable parameter is constrained to a boundary. In a mixture test, the added component can enter at many possible values of \(\eta\). The likelihood-ratio statistic can benefit from choosing whichever value fits the observed sample best. Its null distribution must account for that search.

From a Local Score to a Supremum

The likelihood-ratio statistic is \(T_n=2\{\sup_{\alpha,\eta,\beta}\ell_n(\alpha,\eta,\beta)-\sup_{\beta}\ell_n(0,\eta,\beta)\}\), where \(\ell_n\) is the log likelihood. The second supremum does not depend on \(\eta\), because \(\eta\) disappears under \(H_0\). For a fixed candidate value of \(\eta\), the mixture weight is constrained by \(\alpha\geq 0\), so its local fitted value cannot move in a negative direction.

The following derivation describes the standard local approximation. Assume independent observations from the null baseline; a smooth, identifiable baseline parameter \(\beta\); finite, positive score variances; and sufficient regularity for a uniform Gaussian limit over the allowed values of \(\eta\). Also assume the mixture model has no additional singularities that invalidate the local expansion. These qualifications matter: mixture models can violate regularity in other ways, so the formula is not universal for every mixture.

For each fixed \(\eta\), the score for increasing \(\alpha\) at zero is based on \(s_\eta(x)=f_1(x;\eta)/f_0(x;\beta)-1\), when this density ratio is well-defined and has finite second moment under the null. If \(\beta\) is estimated, the score must be adjusted for the baseline parameter. Write \(\widetilde{s}_\eta\) for the efficient score: the component of \(s_\eta\) remaining after its linear projection onto the baseline score has been removed. Let \(J_\eta=E_0[\widetilde{s}_\eta^2]>0\), and define the standardized efficient score \(Z_n(\eta)=\sum_{i=1}^n\widetilde{s}_\eta(X_i)/\sqrt{nJ_\eta}\).

Here \(\mathcal{H}\) is the allowed set of component parameters, and \(G(\eta)\) is a mean-zero, unit-variance Gaussian process. Its correlation function is the correlation between the efficient scores at different values of \(\eta\): \(\operatorname{Corr}(G(\eta_1),G(\eta_2))= E_0[\widetilde{s}_{\eta_1}\widetilde{s}_{\eta_2}]/\sqrt{J_{\eta_1}J_{\eta_2}}\). The supremum appears because the alternative selects the best-fitting \(\eta\); the positive part appears because \(\alpha\) cannot be negative. This explains why the limiting distribution depends on the range of \(\eta\) and the dependence among its score directions.

Finite Candidate Components

A useful special case restricts \(\eta\) to \(K\) candidate values. If the corresponding limiting scores \(Z_1,\ldots,Z_K\) are independent standard normal variables, then the limiting statistic is \(T=\max_{1\leq j\leq K}\{\max(0,Z_j)\}^2\). For \(t\geq0\), the event \(T\leq t\) occurs exactly when every score is at most \(\sqrt{t}\). Independence therefore gives

where \(\Phi\) is the standard normal cumulative distribution function. In particular, \(P(T=0)=2^{-K}\): all \(K\) scores must be nonpositive. If the scores are correlated, the product formula does not apply. The joint Gaussian distribution, including its correlations, determines the calibration.

Common Exam Trap: Reusing a Boundary Mixture Automatically

A frequent error is to see \(\alpha=0\) at the edge of its parameter space and immediately use the single-direction boundary distribution. That calibration is appropriate only when the local alternative has one relevant, identifiable score direction under the null, along with the required regularity conditions. When \(\eta\) is absent under the null, the alternative searches across score directions, and the limiting statistic is a supremum.

For example, at \(T=2.25\), the wrong single-direction boundary calculation gives \(1-\Phi(1.5)\approx0.0668\). With two independent candidate component values, the correct calculation is \(1-\Phi(1.5)^2\approx0.1292\). The first calculation understates the tail probability because it omits the search over candidates. Counting \(\alpha\) and \(\eta\) and using a chi-square distribution with two degrees of freedom is not a general fix; \(\eta\) is not identified by the null, so the regular Wilks degrees-of-freedom argument does not apply.

For a continuous set of candidate values, there is generally no finite-\(K\) product formula. Calibration requires the covariance structure of the Gaussian score process and the allowed parameter set. The key exam decision is therefore made before looking up a reference distribution: determine whether the nuisance parameter is identified under the null and whether the alternative contributes one direction or a search over directions.