Let $X_1,...,X_n\in R^p$ be i.i.d. with density, $$f_{\mathbf{\theta}}(\mathbf{x})=c(a)exp(-|\mathbf{x-\theta}|^a), \mathbf{\theta}\in R^p, a\geq 1$$$$f_{\mathbf{\theta}}(\mathbf{x})=c(a)\exp(-|\mathbf{x-\theta}|^a), \mathbf{\theta}\in \mathbb R^p, a\geq 1$$ where $c^{-1}(a)=\int_{R^p}exp(-|\mathbf{x}|^a)d\mathbf{x}$$$c^{-1}(a)=\int_{R^p}\exp(-|\mathbf{x}|^a)d\mathbf{x}$$ and $|.|$ is the Euclidean norm.
I want to show that if $a>1$, the MLE$\mathbf{\hat{\theta}}$ exists and is unique.
Corollary on existence and uniqueness of MLE: Suppose that a rank $k$ canonical exponential family distribution has generator $(\mathbf{T},h)$ and an open natural parameter space $\epsilon$. If the probability distribution of $\mathbf{T(X)}$ has a convex support $C_T$, then for the data vector $\mathbf{X}$ observed as $\mathbf{x}$, The MLE exists and is unique iff $\mathbf{T(x)}\in C_T\backslash > \partial C_T$$\mathbf{T(x)}\in C_T\backslash \partial C_T$, the interior of $C_T$. In other words:$$P(\mathbf{c}^T\mathbf{T(X)}>\mathbf{c}^T\mathbf{T(x)})>0$$ for any $\mathbf{c}\neq \mathbf{0}$.
An alike question: Existence and uniqueness of MLE
Any helps would be appreciated!
