Suppose $X$ follows a multinomial distribution $\text{Multi}(n, \pi)$ where $\pi=(\pi_1,...,\pi_K)'$ where $\pi_k > 0$ for all $k$. Let $\theta=(\theta_1,...,\theta_q)$ be a parameter vector with $q < K-1$. Let $h : \mathbb{R}^{q} \rightarrow \mathbb{R}^{K}$ be a known re-parameterization function such that $\pi=h(\theta)$. Assume that the parameter space of $\theta$ is an open set. Let $\hat{\theta}$ be the MLE of $\theta$. Assume that the re-parameterization function $h$ is chosen so that $\hat{\theta}$ is unique and it exists.
My question is how should I derive the limiting distribution of $\sqrt{n}(\hat{\theta}-\theta)$?
I know that under regularity conditions the limiting distribution is $N(0,[I(\theta)]^{-1})$. But the real question is that I cannot frame the likelihood function for $\theta$. My main problem is that the transformation is from $\pi \rightarrow \theta$ but $\theta$ has a lesser dimension than $\pi$. So, I think the Jacobian transformation technique doesn't work here.