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Some of the questions concerning the uniqueness of the maximum likelihood estimates for the parameters in the Weibull distribution are considered for both censored and noncensored samples. In some cases answers previously known are reviewed, while new results are presented for some other cases. In particular, it is shown that with the shape parameter known, the maximum likelihood estimates of the location and scale parameters exist and are unique. Also, the case in which all three parameters are unknown is examined in detail.
Rockette et al. (1974) studied this question.