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

摘要

The problem of nonlinear weighted least squares fitting of the three-parameter Weibull distribution to the given data (w(i), t(i), y(i)), i = 1,..., n, is considered. The part w(i) > 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 < t(1) < t(2) < ... < t(n) and (ii) 0 < y(1) < y(2) < ... < y(n) < 1. To support this, an illustrative numerical example is given.