Correction to: Jayakumar, K., & Sankaran, K. K. (2019). Discrete Linnik Weibull distribution. Communications in Statistics—Simulation and Computation, 48 (10): 3092–117

Abstract

In the article “Discrete Linnik Weibull distribution”, the authors used the method of maximum likelihood to estimate the unknown parameters of the distribution. They made some mistakes in the estimation procedure and section 7. We present the corrections.

Keywords:

• Discrete Linnik distribution
• Maximum likelihood
• Weibull distribution

This article refers to:

Discrete Linnik Weibull distribution

1. Introduction

The partial derivative $\frac{\partial \hspace{0.17em}\text{log}\hspace{0.17em}L}{\partial \beta }$ in Equation (29) is not correct. The log-likelihood function is $$\begin{array}{cc}\hspace{0.17em}\text{log}\hspace{0.17em}L\hfill & =n\hspace{0.17em}\text{log}(\alpha )+n\hspace{0.17em}\text{log}(\nu )-n\hspace{0.17em}\text{log}(\beta )\hfill \\ \hfill & \hspace{1em}+n\nu \hspace{0.17em}\text{log}(p)+n\hspace{0.17em}\text{log}(1-p)-n\hspace{0.17em}\text{log}(1-p^{\nu })\hfill \\ \hfill & \hspace{1em}+(\beta -1)\sum _{i=1}^n\hspace{0.17em}\text{log}(x_i)-\sum _{i=1}^n{x}_i^{\beta }+(\alpha -1)\sum _{i=1}^n\hspace{0.17em}\text{log}(1-e^{-x_i^{\beta }})\hfill \\ \hfill & \hspace{1em}-(\nu +1)\sum _{i=1}^n\hspace{0.17em}\text{log}(p+(1-p){(1-e^{-x_i^{\beta }})}^{\alpha }).\hfill \end{array}$$ Differentiating with respect to β gives $$\frac{\partial \hspace{0.17em}\text{log}\hspace{0.17em}L}{\partial \beta }=\frac{n}{\beta }+\sum _{i=1}^n\hspace{0.17em}\text{log}(x_i)-\sum _{i=1}^n{x}_i^{\beta }\hspace{0.17em}\text{log}(x_i)A_i$$ where $$A_i=1-\frac{(\alpha -1)e^{-x_i^{\beta }}}{1-e^{-x_i^{\beta }}}+\frac{\alpha (\nu +1)(1-p)e^{-x_i^{\beta }}{(1-e^{-x_i^{\beta }})}^{\alpha -1}}{p+(1-p){(1-e^{-x_i^{\beta }})}^{\alpha }}.$$ Moreover, in the proof of Theorem 6.4, the correct expression for $\frac{\partial g_4(\beta ;\alpha ,\nu ,p,1,x)}{\partial \beta }$ is $$\begin{array}{cc}\frac{\partial g_4(\beta ;\alpha ,\nu ,p,1,x)}{\partial \beta }\hfill & =-\frac{n}{{\beta }^2}+\sum _{i=1}^n{x}_i^{\beta }\hspace{0.17em}\text{log}{(x_i)}^2\hfill \\ \hfill & \hspace{1em}-\sum _{i=1}^n\frac{(\alpha -1)x_i^{\beta }\hspace{0.17em}\text{log}{(x_i)}^2{e}^{-x_i^{\beta }}}{1-e^{-x_i^{\beta }}}\left[1-x_i^{\beta }-\frac{x_i^{\beta }{e}^{-x_i^{\beta }}}{1-e^{-x_i^{\beta }}}\right].\hfill \\ \hfill & \hspace{1em}-\sum _{i=1}^n\frac{\alpha (\nu +1)(1-p)x_i^{\beta }\hspace{0.17em}\text{log}{(x_i)}^2{e}^{-x_i^{\beta }}}{p+(1-p){(1-e^{-x_i^{\beta }})}^{\alpha }}{(1-e^{-x_i^{\beta }})}^{\alpha -2}\hfill \\ \hfill & \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\left[(\alpha -1)x_i^{\beta }{e}^{-x_i^{\beta }}-(1-x_i^{\beta })(1-e^{-x_i^{\beta }})-\frac{\alpha (1-p)x_i^{\beta }{e}^{-x_i^{\beta }}{(1-e^{-x_i^{\beta }})}^{\alpha }}{p+(1-p){(1-e^{-x_i^{\beta }})}^{\alpha }}\right]\hfill \end{array}$$ In Section 7, the authors incorrectly reported the data set from Bekker, Roux, and Mosteit (2000). The data set is

Table

0.047	0.115	0.121	0.132	0.164	0.197	0.203	0.260	0.282	0.296	0.334	0.395
0.458	0.466	0.501	0.507	0.529	0.534	0.540	0.570	0.641	0.644	0.696	0.841
0.863	1.099	1.219	1.271	1.326	1.447	1.485	1.553	1.581	1.589	2.178	2.343
2.416	2.444	2.825	2.830	3.578	3.658	3.743	3.978	4.033

The values in the Table 4 are correct.