(i) State Wilks' likelihood ratio test of the null hypothesis $H_{0}: \theta \in \Theta_{0}$ against the alternative $H_{1}: \theta \in \Theta_{1}$, where $\Theta_{0} \subset \Theta_{1}$. Explain when this test may be used.

(ii) Independent identically-distributed observations $X_{1}, \ldots, X_{n}$ take values in the set $S=\{1, \ldots, K\}$, with common distribution which under the null hypothesis is of the form

$$P\left(X_{1}=k \mid \theta\right)=f(k \mid \theta) \quad(k \in S)$$

for some $\theta \in \Theta_{0}$, where $\Theta_{0}$ is an open subset of some Euclidean space $\mathbb{R}^{d}$, $d<K-1$. Under the alternative hypothesis, the probability mass function of the $X_{i}$ is unrestricted.

Assuming sufficient regularity conditions on $f$ to guarantee the existence and uniqueness of a maximum-likelihood estimator $\hat{\theta}_{n}\left(X_{1}, \ldots, X_{n}\right)$ of $\theta$ for each $n$, show that for large $n$ the Wilks' likelihood ratio test statistic is approximately of the form

$$\sum_{j=1}^{K}\left(N_{j}-n \hat{\pi}_{j}\right)^{2} / N_{j}$$

where $N_{j}=\sum_{i=1}^{n} I_{\left\{X_{i}=j\right\}}$, and $\hat{\pi}_{j}=f\left(j \mid \hat{\theta}_{n}\right)$. What is the asymptotic distribution of this statistic?