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We attempt to investigate a two-dimensional Gauss-Kuzmin theorem forR\'enyi-type continued fraction expansions. More precisely speaking, our focusis to obtain specific lower and upper bounds for the error term consideredwhich imply the convergence rate of the distribution function involved to itslimit. To achieve our goal, we exploit the significant properties of thePerron-Frobenius operator of the R\'enyi-type map under its invariant measureon the Banach space of functions of bounded variation. Finally, we give somenumerical calculations to conclude the paper.

Two asymptotic distributions related to R\'enyi-type continued fraction expansions