This work is about the absolute stability studies of closed-loop data-driven models with sector nonlinearities based on Koopman operator theory with data-dependent bounds of the estimation. This method leads to a class of stability analysis where the accuracy of the estimation depends on the non-asymptotic convergence of the error estimation based on Dynamic Mode Decomposition (DMD). Given this framework, we yield a quadratic Lyapunov analysis using the Strictly Positive Real Lemma (SPR) and Circle criterion to study the convergence of the trajectories to the equilibrium point. In the last part, we show numerical example results depicting the application of the proposed methodology.