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Let $\nabla $ be a linear connection on an $2n$-dimensional almostanti-Hermitian manifold $M$\ equipped with an almost complex structure $J$, apseudo-Riemannian metric $g$ and the twin metric $G=g\circ J$. In this paper,we first introduce three types of conjugate connections of linear connectionsrelative to $g$, $G$ and $J$. We obtain a simple relation among curvaturetensors of these conjugate connections. To clarify relations of these conjugateconnections, we prove a result stating that conjugations along with an identityoperation together act as a Klein group. Secondly, we give some resultsexhibiting occurrences of Codazzi pairs which generalize parallelism relativeto $\nabla $. Under the assumption that $(\nabla ,J)$ being a Codazzi pair, \we derive a necessary and sufficient condition the almost anti-Hermitianmanifold $(M,J,g,G)$ is an anti-K\"{a}hler relative to a torsion-free linearconnection $\nabla $. Finally, we investigate statistical structures on $M$under $\nabla $ ($\nabla $ is a $J-$invariant torsion-free connection).

Notes concerning Codazzi pairs on almost anti-Hermitian manifolds