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In this paper, we propose a real-time tone-mapping scheme for backward compatible high dynamic range (HDR) video compression. The appropriate choice of a tone-mapping operator (TMO) can significantly improve the HDR quality reconstructed from a low dynamic range (LDR) version. We develop a statistical model that approximates the mean square error (MSE) distortion resulting from the combined processes of tone-mapping and compression. Using this model, we formulate a numerical optimization problem to find the tone-curve that minimizes the expected MSE in the reconstructed HDR sequence. We then simplify the developed model in order to reduce the computational complexity of the optimization problem to a closed-form solution. Performance evaluations show that the proposed methods provide superior performance in terms of HDR MSE and SSIM compared to existing tone-mapping schemes. It is also shown that the LDR image quality resulting from the proposed methods matches that produced by perceptually-based TMOs.