 Solution

The unrestricted likelihood is maximized at the empirical proportions, while the null likelihood is maximized at ${\pi }_i(\hat{\theta })$. Twice the log of their ratio isLarge values mean the fitted null likelihood is much smaller than the unrestricted maximum, so rejection is in the upper tail.

For a regular identifiable scalar parameter lying in the interior, positive true cell probabilities, and sufficiently large expected counts, Wilks theorem gives an approximate chi-squared distribution with $(k-1)-1=k-2$ degrees of freedom under the null. The unrestricted probability vector has dimension $k-1$ and the fitted null has dimension one. At significance level $\alpha$, reject if $\Lambda (\hat{\theta })>F_{k-2}^{-1}(1-\alpha )$, or report the approximate p-value $1-F_{k-2}(\Lambda )$.

Cells with very small expected counts may require a prespecified pooling scheme or simulation calibration; the asymptotic reference is not an exact finite-sample law. For a regular null with $d$ fitted parameters, the corresponding degrees of freedom are $k-1-d$.