Let $\mathbf{x} = [x_1,\ldots,x_N] \in \mathbb{C}^N$ be a set of complex-valued data and let $p(\mathbf{x};\alpha)$ their joint probability density function parametrized by the unknown complex scalar $\alpha \in \mathbb{C}$. Suppose now to have the following composite hypothesis testing problem:
\begin{equation}
H_0: \alpha = 0\quad \mathrm{vs.}\quad H_1: \alpha \neq 0.
\end{equation}
I'm trying to derive a Wald-type test directly on the complex field, but I'm having some problems. In particular, which is the asymptotic distribution of a "complex" Wald test?
Can anyone suggest me some works on complex-valued hypothesis testing problems (and, in particular on complex-valued Wald statistics) ?
