On the existence of the nonlinear weighted least squares estimate for a three-parameter Weibull distribution

The problem of nonlinear weighted least squares fitting of the three-parameter Weibull distribution to the given data (wi,ti,yi)(wi,ti,yi), i=1,…,ni=1,…,n, is considered. The part wi>0wi>0 of the data stands for the data weights. It is shown that the best least squares estimate exists provided that the data satisfy just the following two natural conditions: (i) 0