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In this work, we consider the two-way relay network (TWRN) where two terminals exchange their information through a relay node in a bi-directional manner and study the training-based channel estimation under the amplify-and-forward (AF) relay scheme. We propose a two-phase training protocol for channel estimation: in the first phase, the two terminals send their training signals concurrently to the relay; and in the second phase, the relay amplifies the received signal and broadcasts it to both terminals. Each terminal then estimates the channel parameters required for data detection. First, we assume the channel parameters to be deterministic and derive the maximum-likelihood (ML) -based estimator. It is seen that the newly derived ML estimator is nonlinear and differs from the conventional least-square (LS) estimator. Due to the difficulty in obtaining a closed-form expression of the mean square error (MSE) for the ML estimator, we resort to the Crameacuter-Rao lower bound (CRLB) on the estimation MSE for design of optimal training sequence. Secondly, we consider stochastic channels and focus on the class of linear estimators. In contrast to the conventional linear minimum-mean-square-error (LMMSE) -based estimator, we introduce a new type of estimator that aims at maximizing the effective receive signal-to-noise ratio (SNR) after taking into consideration the channel estimation errors, thus referred to as the linear maximum SNR (LMSNR) estimator. Furthermore, we prove that orthogonal training design is optimal for both the CRLB- and the LMSNR-based design criteria. Finally, simulations are conducted to corroborate the proposed studies.