Correction

Article title: The Weibull Fréchet distribution and its applications

Authors: Ahmed Z. Affify, Haitham M. Yousof, Gauss M. Cordeiro, Edwin M. M. Ortega & Zohdy M. Nofal

Journal: Journal of Applied Statistics

Bibliometrics: Volume 43, Number 14, pages 2608–2626

DOI: http://dx.doi.org/10.1080/02664763.2016.1142945

After reviewing calculations related to the likelihood function and the score vector, the authors wish to correct some typographical errors in their paper which do not invalidate the methodological and applied contributions of their paper.

‘Estimation’ Section 6 and Equation (17) have now been corrected to the following:

The MLEs of the parameters of the WFr $(\alpha ,\beta ,a,b,x)$ model is now discussed. Let $\mathbf{x}=(x_1,\dots ,x_n)$ be a random sample of this distribution with unknown parameter vector $\theta =(\alpha ,\beta ,a,b{)}^{⊺}$.

The log-likelihood function for $\theta$, say $\ell =\ell (\theta )$, is given by

$$\begin{array}{rl}\ell & =n(\log a+\log b+\log \beta +\log {\alpha }^{\beta })-(\beta +1)\sum _{i=1}^n\log x_i-b\sum _{i=1}^n{\left(\frac{\alpha }{x_i}\right)}^{\beta }\\ & -(b+1)\sum _{i=1}^n\log (1-s_i)-a\sum _{i=1}^n{\left(\frac{s_i}{1-s_i}\right)}^b,\end{array}$$

where $s_i=\exp [-{\left(\frac{\alpha }{x_i}\right)}^{\beta }]$.

The score vector is given by $\mathbf{U}(\theta )=\frac{\mathrm{\partial }\ell }{\mathrm{\partial }\theta }={(\frac{\mathrm{\partial }\ell }{\mathrm{\partial }\alpha },\frac{\mathrm{\partial }\ell }{\mathrm{\partial }\beta },\frac{\mathrm{\partial }\ell }{\mathrm{\partial }a},\frac{\mathrm{\partial }\ell }{\mathrm{\partial }b})}^{⊺}$.

Let $z_i={\left(\frac{\alpha }{x_i}\right)}^{\beta }\log \left(\frac{\alpha }{x_i}\right)$. Then,

$$\begin{array}{c}\frac{\mathrm{\partial }\ell }{\mathrm{\partial }\alpha }=\frac{n\beta }{\alpha }-\frac{b\beta }{{\alpha }^{1-\beta }}\sum _{i=1}^n{x}_i^{-\beta }-\frac{(b+1)\beta }{{\alpha }^{1-\beta }}\sum _{i=1}^n\frac{s_i{x}_i^{-\beta }}{1-s_i}+\frac{ab\beta }{{\alpha }^{1-\beta }}\sum _{i=1}^n\frac{s_i^{-1}{x}_i^{-\beta }}{(s_i^{-1}-1{)}^{b+1}},\\ \frac{\mathrm{\partial }\ell }{\mathrm{\partial }\beta }=\frac{n}{\beta }+n\log \alpha -\sum _{i=1}^n\log x_i-b\sum _{i=1}^n{z}_i-(b+1)\sum _{i=1}^n\frac{s_i{z}_i}{1-s_i}+\sum _{i=1}^n\frac{ab{s}_i^{-1}{z}_i}{(s_i^{-1}-1{)}^{b+1}},\\ \frac{\mathrm{\partial }\ell }{\mathrm{\partial }a}=\frac{n}{a}-\sum _{i=1}^n(s_i^{-1}-1{)}^{-b}\end{array}$$

and

$$\frac{\mathrm{\partial }\ell }{\mathrm{\partial }b}=\frac{n}{b}+\sum _{i=1}^n\log s_i-\sum _{i=1}^n\log (1-s_i)-a\sum _{i=1}^n(s_i^{-1}-1{)}^{-b}\log (s_i^{-1}-1).$$

The authors, the Editor-in-Chief and the Publisher would like to thank the authors of the Comment (http://dx.doi.org/10.1080/02664763.2021.1973388) for bringing their concerns to our attention.