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A two-way relay non-orthogonal multiple access (TWR-NOMA) system is investigated, where two groups of NOMA users exchange messages with the aid of one half-duplex decode-and-forward relay. Since the signal-plus-interference-to-noise ratios of NOMA signals mainly depend on effective successive interference cancellation (SIC) schemes, imperfect SIC (ipSIC), and perfect SIC (pSIC) are taken into account. In order to characterize the performance of TWR-NOMA systems, we first derive closed-form expressions for both exact and asymptotic outage probabilities of NOMA users' signals with ipSIC/pSIC. Based on the derived results, the diversity order and throughput of the system are examined. Then, we study the ergodic rates of users' signals by providing the asymptotic analysis in high signal-to-noise ratio (SNR) regimes. Finally, numerical simulations are provided to verify the analytical results and show that …