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This study deals with the performance analysis and power allocation of a two-way amplify-and-forward relay system with channel estimation errors. Exact closed-form expressions for outage probability and average symbol error rate (SER) are first presented. To provide more insights, their closed-form asymptotic expressions are then obtained. It is shown that the presence of channel estimation error causes outage probability and average SER maintain a fixed level even when a noiseless channel is adopted. These results are applied further to study the optimal power allocation problem for each node. From the perspective of service quality, rather than minimise the outage probability as much as possible, the goal is to use the minimum energy consumed to satisfy the traffic requirements and thereby conserve the energy resource. Furthermore, based on the optimal solutions to power allocation, the optimal relay location is investigated, which indicates that the system has higher energy efficiency with the relay located around the middle point of the two end nodes for any asymmetric traffic requirement. The simulation results verify that the derived outage probability and average SER expressions are accurate and highlight the effect of power allocation under various traffic requirements and channel estimation errors.