I am trying to understand the Corollary 2.3.2 from Bickel's book mathematical statistics.
The corollary says
Consider the exponential family$$p(x,\theta)=h(x)\exp\left\{\sum_{j=1}^kc_j(\theta)T_j(x)-B(\theta)\right\},x\in\mathcal{X},\theta\in\Theta.$$ Let $C^0$ denote the interior of the range of $(c_1(\theta),c_2(\theta),\dots,c_k(\theta))^T$ and let $x$ be the observed data. If the equations $$E_\theta T_j(X)=T_j(x),j=1,\dots,k$$ have a solution $C(\hat{\theta})\in C^0,$ then it is the unique MLE of $\theta$.
I think the conclusion of this corollary is useful, as it is not easy to show the existence and uniqueness of a curved exponential family in general. However, the book does not given any example of using this corollary.
I am trying to understand this corollary. Can someone give an example showing the usefulness of it?
After some exploration, I think one example is that when we proving the uniqueness of the MLE of the bivariate normal distribution. In the bivariate normal distribution, we have five parameters $\theta = (\mu_1,\mu_2,\sigma_1, \sigma_2,\rho)\in({\mathbb{R},\mathbb{R},\mathbb{R}^+,\mathbb{R}^+,(0,1)})=\Theta$. We could use the Hessian of the p.d.f being negative definite to show that $\hat{\theta}_{MLE}$ is unique. But calculating the second derivative with respect to five parameters is a pain. However, using the corollary, if we could show that $c(\hat{\theta}_{MLE})$ lays in the interior of $c(\Theta)$, we get the uniqueness of the $\hat{\theta}_{MLE}$.
Please note however, this corollary cannot be used for cases like $\mathcal{N}(\theta,\theta^2)$, from which $c(\Theta)$ is a parabola in $\mathbb{R}^2$, that does have an "interior" of a ball in $\mathbb{R}^2$. Thus, this corollary does not apply.