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This paper is concerned with power allocation (PA) among the source and the relays of wireless relay networks to minimize the mean-square error (MSE) of the channel estimation when distributed space-time coding (DSTC) is applied. The optimal PA scheme is numerically obtained by means of geometric programming for the least-squares (LS) estimator, and a closed-form near-optimal PA scheme is also suggested. The impact of imperfect channel estimation on the error performance of DSTC is analyzed for both the LS and linear minimum MSE (LMMSE) channel estimators. It is proved that mismatched decoding of DSTC is able to achieve the same diversity order as coherent decoding of DSTC. Furthermore, when the closed-form PA obtained in the training phase is applied to the transmission phase, mismatched decoding is able to achieve a significant coding gain over the equal-PA scheme (which assigns half of the total power to the source and equally shares the other half to all the relays).