Estimation and filtering form an important component of most modern control systems. Techniques such as extended Kalman filters and particle filters have been successfully utilized for estimation in many different applications. Integrators derived from discrete mechanics possess desirable numerical properties such as stable long-time energy behavior, exact constraint satisfaction, and accurate statistical calculations. In the present work, we leverage these features by utilizing a variational integrator derived from discrete mechanics within extended Kalman filters and particle filters. By filtering real experimental data from the nonlinear, underactuated planar crane problem we demonstrate that the linearizations available through the discrete mechanics framework increase the accuracy of uncertainty estimates provided by an extended Kalman filter, especially when operating at low frequencies. Additionally, we illustrate situations where particle filter performance is increased through the statistics-preserving properties provided by the variational integrator.