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Multimodal Sentiment Analysis (MSA) has attracted widespread research attention recently. Most MSA studies are based on the assumption of modality completeness. However, many inevitable factors in real-world scenarios lead to uncertain missing modalities, which invalidate the fixed multimodal fusion approaches. To this end, we propose a Unified multimodal Missing modality self-Distillation Framework (UMDF) to handle the problem of uncertain missing modalities in MSA. Specifically, a unified self-distillation mechanism in UMDF drives a single network to automatically learn robust inherent representations from the consistent distribution of multimodal data. Moreover, we present a multi-grained crossmodal interaction module to deeply mine the complementary semantics among modalities through coarse- and fine-grained crossmodal attention. Eventually, a dynamic feature integration module is introduced to enhance the beneficial semantics in incomplete modalities while filtering the redundant information therein to obtain a refined and robust multimodal representation. Comprehensive experiments on three datasets demonstrate that our framework significantly improves MSA performance under both uncertain missing-modality and complete-modality testing conditions.