<p>This paper focuses on using high-range resolution radar for tracking and classification of extended objects, taking into account unknown measurement noise. While most existing methods only address tracking, target classification is also crucial in many scenarios. In particular, for extended objects, tracking and classification should be considered together as they can impact each other. Additionally, in most real-world nonlinear state estimation problems, the measurement noise is unknown. To solve the above problems, an extended object modeling and multiple model tracking algorithm that combines statistical linearization variational Bayesian with down-range extent and cross-range extent is proposed. By integrating the shape information of the extended object, a method based on support functions or extended Gaussian image is proposed to re-model the system dynamic model of smooth and non-smooth extended objects, respectively. Based on the multiple model strategy, an extended target tracking and classification algorithm is derived to jointly obtain estimates of kinematic states and target extension in each class, as well as the posterior probability of the target class. To further improve accuracy, the paper introduces a statistical linearized variational Bayesian estimator to adaptively estimate the variance of unknown measurement noise in real-time. The effectiveness of the proposed approach is illustrated through simulation results.</p>