This note provides an effective bound in the Gauss-Kuzmin-L\'evy problem for some Gauss type shifts associated with nearest integer continued fractions, acting on the interval $I_0=[0,\frac{ 1}{ 2}]$ or $I_0=[-\frac{ 1}{ 2},\frac{ 1}{ 2}]$. We prove asymptotic formulas $\lambda (T^{ -n}I) =\mu(I)(\vert I_0 \vert +O(q^n))$ for such transformations $T$, where $\lambda$ is the Lebesgue measure on $\mathbb R$, $\mu$ the normalized $T$-invariant Lebesgue absolutely continuous measure, $I$ subinterval in $I_0$, and $q=0.288$ is smaller than the Wirsing constant $q_W=0.3036\ldots$
Categories: math.NT math.DS