We investigate multiple-input multiple-output broadcast channel with channel distribution information at the transmitter. We consider the fading-paper model with Nt transmit and Nr receive antennas. Near-optimal methods for designing the inflation factor matrix (choice of auxiliary random variable) to maximize the linear assignment capacity in a general fading channel model are proposed. Further, it is proved that in the special case of Nt <= Nr, the proposed inflation factor matrix has the similar structure and achieves the similar interference elimination effect as dirty-paper coding. Based on the obtained inflation factor matrix, we provide efficient approaches to evaluate the linear assignment achievable rate region for some popular statistical fading channel models. Monte-Carlo simulation results are provided to demonstrate the effectiveness of our proposed schemes.