Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 27I (Principles of Statistics)

The likelihood-ratio statistic for nested parameter spaces isUnder the null, Wilks theorem gives $W_n\Rightarrow {\chi }_{d_1-d_0}^2$ when the full model and null submodel are smooth and identifiable, the true parameter is an interior regular point, Fisher information is nonsingular in the identifiable directions, and the usual differentiability and likelihood-asymptotic conditions hold. A level-$\alpha$ test rejects for $W_n$ exceeding the upper $\alpha$ quantile of that chi-square law. The approximation is for large samples; boundary parameters, singularities or nonidentifiability can give a different limiting distribution.

The unrestricted categorical MLE is ${\tilde{\pi }}_j=N_j/n$, so the multinomial combinatorial factor cancels from the likelihood ratio andAssume the true cell probabilities are positive and the null has identifiable dimension $d$ with full-rank information. With probability tending to one, all $N_j$ are positive. Put ${\delta }_j=n{\hat{\pi }}_j-N_j$. Regular estimation gives ${\delta }_j=O_p(\sqrt{n})$ and $N_j$ of order $n$. Expanding $-2N_j\log (1+{\delta }_j/N_j)$ yieldsThe linear terms cancel exactly because ${\sum }_j{\delta }_j=0$. ThereforeThe full probability simplex has dimension $K-1$, explaining the degrees of freedom. Existence and uniqueness of an MLE alone do not supply all Wilks regularity assumptions; positivity, smoothness and the local rank condition are also needed. The displayed approximation is not a finite-sample prescription for division by a zero observed count.