  After providing an overview of $\theta$-expansions introduced by Chakraborty
and Rao, we focus on the Gauss-Kuzmin problem for this new transformation.
Actually, we complete our study on these expansions by proving a
two-dimensional Gauss-Kuzmin theorem. More exactly, we obtain such a theorem
related to the natural extension of the associated measure-dynamical system.
Finally, we derive explicit lower and upper bounds of the error term which
provide interesting numerical calculations for the convergence rate involved.
