Recent advancements have focused on developing automated diagnostic systems for detecting schizophrenia (SZ) and bipolar disorder using medical imaging techniques, particularly MRI. However, acquiring MRI scans is costly and requires patients to remain still, which poses significant challenges for individuals with SZ and bipolar disorder. Additionally, many developing countries face a shortage of mental health professionals, further complicating the diagnosis of these conditions. Research has also consistently highlighted motor abnormalities in SZ and bipolar disorder since their earliest descriptions. Notably, recent studies have identified substantial statistical differences in handwriting features between patients with these disorders and healthy individuals. Moreover, it is well-established that machine learning models trained on imbalanced datasets often exhibit biased performance. To address these challenges, we propose a novel three-stage machine learning framework called SRF-SMOTE-NET, which integrates statistically robust features method and oversampling techniques with neural network. In the first stage, we apply the Kruskal-Wallis statistical test to analyze kinematic, complexity, and geometrical features from the handwriting data and identify statistically significant features, termed Statistically Robust Features (SRF). The second stage involves applying the SMOTE technique to address class imbalance, ensuring a fair training process. Finally, in the third stage, we develop a neural network (NET) model to perform unbiased classification. The proposed model is validated using a handwriting dataset, achieving an overall classification accuracy of 96%, and Mathews correlation co-efficient of 0.919.