For the basic maximum likelihood estimating function of the two parameters Weibull distribution, a simple proof on its global monotonicity is given to ensure the existence and uniqueness of its solution. The boundary of the function's first-order derivation is defined based on its scale-free property. With a bounded derivation, the possible range of the root of this function can be determined. A novel root-finding algorithm employing these established results is proposed accordingly, its convergence is proved analytically as well. Compared with other typical algorithms for this problem, the proposed algorithm has a higher efficiency, its performance is also demonstrated by numerical experiments.
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