  The main aim of this paper is to extend the semiparametric inference
methodology, recently investigated for Real Elliptically Symmetric (RES)
distributions, to Complex Elliptically Symmetric (CES) distributions. The
generalization to the complex field is of fundamental importance in all
practical applications that exploit the complex representation of the acquired
data. Moreover, the CES distributions has been widely recognized as a valuable
and general model to statistically describe the non-Gaussian behaviour of
datasets originated from a wide variety of physical measurement processes. The
paper is divided in two parts. In the first part, a closed form expression of
the constrained Semiparametric Cram\'{e}r-Rao Bound (CSCRB) for the joint
estimation of complex mean vector and complex scatter matrix of a set of
CES-distributed random vectors is obtained by exploiting the so-called
\textit{Wirtinger} or $\mathbb{C}\mathbb{R}$-\textit{calculus}. The second part
deals with the derivation of the semiparametric version of the Slepian-Bangs
formula in the context of the CES model. Specifically, the proposed
Semiparametric Slepian-Bangs (SSB) formula provides us with a useful and
ready-to-use expression of the Semiparametric Fisher Information Matrix (SFIM)
for the estimation of a parameter vector parametrizing the complex mean and the
complex scatter matrix of a CES-distributed vector in the presence of unknown,
nuisance, density generator. Furthermore, we show how to exploit the derived
SSB formula to obtain the semiparametric counterpart of the Stochastic CRB for
Direction of Arrival (DOA) estimation under a random signal model assumption.
Simulation results are also provided to clarify the theoretical findings and to
demonstrate their usefulness in common array processing applications.
