This paper presents a nonlinear numerical observer for accurate position, velocity and attitude (PVA) estimation including the accelerometer bias and gyro bias estimation. The Moving Horizon Observer (MHO) processes the accelerometer, gyroscope and magnetometer measurements from the Inertial Measurement Unit (IMU) and the position and velocity measurements from the Global Navigation Satellite System (GNSS). The nonlinear measurement equations with the rotation matrix, expressed through the quaternion parametrization, in combination with the state-space rigid body kinematic model of translational and rotational motion is the subject of optimization defined on a receding data window. The gradient-based trust-region method is applied to solve the MHO's nonlinear least-squares criterion. The MHO is tesed off-line in the numerical experiment involving the experimental flight data from a light fixed-wing aircraft. This study demonstrates the quality of the MHO computations with the comparison of the reference filter, the multiplicative Extended Kalman Filter (EKF).