In (Dong and Ding, 2006), it has been proposed that channel estimates in quadratic form can be obtained at the base station by sending training sequences to the mobiles where the received signals are forwarded back to the base for channel estimation. In this paper, we first examine the optimal training sequence design for such quadratic channel estimation and then analyze the error bound and statistics of the channel estimates in quadratic form. With the analytical results, two problems for a multiple-input single-output (MISO) antenna system in the downlink are constructed and optimally solved: Power minimization with individual users' 1) worst-case signal-to-interference plus noise ratio (SINR) and 2) average mean-square-error (MSE) constraints, through optimal multiuser MISO beamforming and power allocation.