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The combination of non-orthogonal multiple access (NOMA) and mobile edge computing (MEC) can significantly improve the spectrum efficiency beyond the fifth-generation network. In this paper, we mainly focus on energy-efficient resource allocation for a multi-user, multi-BS NOMA assisted MEC network with imperfect channel state information (CSI), in which each user can upload its tasks to multiple base stations (BSs) for remote executions. To minimize the energy consumption, we consider jointly optimizing the task assignment, power allocation and user association. As the main contribution, with imperfect CSI, the optimal closed-form expressions of task assignment and power allocation are analytically derived for the two-BS case. Specifically, the original formulated problem is nonconvex. We first transform the probabilistic problem into a non-probabilistic one. Subsequently, a bilevel programming method is proposed to derive the optimal solution. In addition, by incorporating the matching algorithm with the optimal task and power allocation, we propose a low complexity algorithm to efficiently optimize user association for the multi-user and multi-BS case. Simulations demonstrate that the proposed algorithm can yield much better performance than the conventional OMA scheme but also the identical results with lower complexity from the exhaustive search with the small number of BSs.