Quantum random number generation attracts considerable attention, since its randomness inherently originates in quantum mechanics, but not mathematical assumptions. Randomness certification, e.g. entropy estimation, becomes a key issue in the context of quantum random number generation protocol. We study a self-testing protocol based on dimension witness, with the assumption of independent devices. It addresses the random number extraction problem in a practical prepare-and-measure scenario with uncharacterized devices. However, the lower bound of min-entropy as a function of dimension witness is not tight in existing works. We present a tighter bound of analytic form, by introducing the Lagrangian multiplier method to closely analyze the optimization problem on average guessing probability. Through simulation, it turns out that a significantly higher random number generation rate can be achieved in practice.

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