Nonlinear weighted least squares estimation of a three-parameter Weibull density with a nonparametric start
The authors consider a three parameter Weibull distribution with a density \(f(x;\alpha,\beta,\eta)\) and wish to estimate its parameters. The main improvement over the previously suggested methods is the proposal to combine parametric and nonparametric estimators in the following way. First, a nonparametric estimator \(\hat f\) is constructed from the data \(\{t_i\}\) and the values of \(y_i=\hat f(t_i)\) are evaluated. Second, the parameters \(\alpha,\beta,\eta\) are estimated as \[ \arg\min_{\alpha>0,\beta\geq0,\eta\geq0}\;\sum_{i=1}^{n}w_i\big[y_i-f(t_i;\alpha,\beta,\eta)\big]^2, \] where \(w_i\) are the data weights describing the assumed relative accuracy of the data. The main result of the paper, Theorem~1, guarantees the existence of such a solution. A small simulation study completes the paper.
three parameter Weibull distribution data fitting
scientific article; zbMATH DE number 5559057
scientific article