We give a new proof of the theorems on the maximum entropy principle in Tsallis statistics. That is, we show that the q-canonical distribution attains the maximum value of the Tsallis entropy, subject to the constraint on the q-expectation value and the q-Gaussian distribution attains the maximum value of the Tsallis entropy, subject to the constraint on the q-variance, as applications of the nonnegativity of the Tsallis relative entropy, without using the Lagrange multipliers method. In

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... n, we define a q-Fisher information and then prove a q-Cramér-Rao inequality that the q-Gaussian distribution with special q-variances attains the minimum value of the q-Fisher information.