Letter to the editor: Correction to “Lindley—exponential distribution: properties and applications”

Abstract

Bhati et al. (Metron 73:335–357, 2015) introduced and studied a new class of distributions. However, some of the data were not entered correctly and, therefore, the results may diverge somewhat from what the authors find in the original paper should the data be corrected. In other words, the results we obtain differ from the original ones.

Appendices

Appendices

A : Mathematica code based on original paper data

$y=\{0.08,2.09,3.48,4.87,6.94,8.66,13.11,23.63,0.2,2.23,0.26,0.31,0.73,0.52,4.98, 6.97,9.02,13.29,$

$0.4,2.26,3.57,5.06,7.09,11.98,4.51,2.07,0.22,13.8,25.74,0.5,2.46,3.64,5.09,7.26, 9.47,14.24,$

$19.13,6.54,3.36,0.82,0.51,2.54,3.7,5.17,7.28,9.74,14.76,26.31,0.81,1.76,8.53, 6.93, 0.62,3.82,$

$5.32,7.32,10.06,14.77,32.15,2.64,3.88,5.32,3.25,12.03,8.65,0.39, 10.34,14.83, 34.26,0.9,2.69,$

$4.18,5.34,7.59,10.66,4.5,20.28,12.63,0.96,36.66,1.05,2.69,4.23,5.41, 7.62,10.75, 16.62,43.01,$

$6.25,2.02,22.69,0.19,2.75,4.26,5.41,7.63,17.12,46.12,1.26,2.83,4.33,8.37,3.36,5.49,0.66,11.25,$

$17.14,79.05,1.35,2.87,5.62,7.87,11.64,17.36,12.02,6.76,0.4,3.02,4.34,5.71,7.93,11.79,18.1,1.46,$

$4.4,5.85,2.02,12.07\};$

$n=\text {Length}[y];$

$\text {For}[i=1,i\le n,i\text {++},\text {Subscript}[x,i]=y[[i]]];$

$\text {LE}=\text {NMaximize}[\{\sum _{i=1}^n \text {Log}\left[ \frac{\theta ^2\lambda e^{-\lambda x_i}\left( 1- e^{-\lambda x_i}\right) {}^{\theta -1}\left( 1-\text {Log}\left[ 1- e^{-\lambda x_i}\right] \right) }{1+\theta }\right] ,\theta>0,\lambda >0\},\{\theta ,\lambda \}];$

$\text {PL}=\text {NMaximize}[\{n \text {Log}\left[ \frac{\alpha \beta ^2}{1+\beta }\right] +\sum _{i=1}^n \text {Log}\left[ \left( 1+x_i^{\alpha }\right) x_i^{\alpha -1}\right] -\sum _{i=1}^n \beta x_i^{\alpha },\alpha>0,\beta >0\},\{\alpha ,\beta \}];$

$L=\text {NMaximize}[\{\sum _{i=1}^n \text {Log}\left[ \frac{ \theta ^2}{1+\theta }\left( 1+x_i\right) e^{-\theta x_i}\right] ,\theta >0\},\{\theta \}];$

$\text {NGLD}=\text {NMaximize}[\{\sum _{i=1}^n \text {Log}\left[ \frac{e^{-\theta x_i}}{1+\theta }\left( \frac{\theta ^{\alpha +1}x_i^{\alpha -1}}{\text {Gamma}[\alpha ]}+\frac{\theta ^{\beta }x_i^{\beta -1}}{\text {Gamma}[\beta ]}\right) \right] ,\alpha>0,\theta>0,\beta >0\},\{\alpha ,\theta ,\beta \}];$

$\nu =3;\rho =2;\eta =1;$

$\text {Ta}=\{\{\text {{``}Distribution{''}}, \text {{``}Parameters Estimates{''}},\text {{``}-LL{''}},\text {{``}AIC{''}},\text {{``}BIC{''}}\},$

$\{\text {{``}LE{''}},\text {LE}[[2]],-\text {LE}[[1]],-2*\text {LE}[[1]]+2*\rho ,-2*\text {LE}[[1]]+\text {Log}[n]*\rho \},$

$\{\text {{``}PL{''}},\text {PL}[[2]],-\text {PL}[[1]],-2*\text {PL}[[1]]+2*\rho ,-2*\text {PL}[[1]]+\text {Log}[n]*\rho \},$

$\{\text {L},L[[2]],-L[[1]],-2*L[[1]]+2*\eta ,-2*L[[1]]+\text {Log}[n]*\eta \},$

$\{\text {{``}NGLD{''}},\text {NGLD}[[2]],-\text {NGLD}[[1]],-2*\text {NGLD}[[1]]+2*\nu ,-2*\text {NGLD}[[1]]+\text {Log}[n]*\nu \}\};$

$\text {MatrixForm}[\text {Ta}]$

The output of the above program code is as follows

$\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \text {Distribution} &{} \text {Parameters Estimates} &{} \text {-LL} &{} \text {AIC} &{} \text {BIC} \\ \text {LE} &{} \{\theta \rightarrow 1.22853,\lambda \rightarrow 0.0962377\} &{} 401.782 &{} 807.564 &{} 813.268 \\ \text {PL} &{} \{\alpha \rightarrow 0.744271,\beta \rightarrow 0.385473\} &{} 402.237 &{} 808.475 &{} 814.179 \\ \text {L} &{} \{\theta \rightarrow 0.21292\} &{} 417.924 &{} 837.848 &{} 840.7 \\ \text {NGLD} &{} \{\alpha \rightarrow 3.56621,\theta \rightarrow 0.145575,\beta \rightarrow 0.983276\} &{} 402.537 &{} 811.074 &{} 819.63 \\ \end{array} \right)$

B : Mathematica Output based on Lee and Wang (2003, p. 231) data

The first command in Appendix A is changed as follows.

$y=\{0.08,0.2,0.4,0.5,0.51,0.81,0.9,1.05,1.19,1.26,1.35,1.4,1.46,1.76,2.02,2.02,2.07,2.09,2.23,$

$2.26,2.46,2.54,2.62,2.64,2.69,2.69,2.75,2.83,2.87,3.02,3.25,3.31,3.36,3.36,3.48,3.52,3.57,$

$3.64,3.7,3.82,3.88,4.18,4.23,4.26,4.33,4.34,4.4,4.5,4.51,4.87,4.98,5.06,5.09,5.17,5.32,5.32,$

$5.34,5.41,5.41,5.49,5.62,5.71,5.85,6.25,6.54,6.76,6.93,6.94,6.97,7.09,7.26,7.28,7.32,7.39,$

$7.59,7.62,7.63,7.66,7.87,7.93,8.26,8.37,8.53,8.65,8.66,9.02,9.22,9.47,9.74,10.06,10.34,10.66,$

$10.75,11.25,11.64,11.79,11.98,12.02,12.03,12.07,12.63,13.11,13.29,13.8,14.24,14.76,14.77,14.83,$

$15.96,16.62,17.12,17.14,17.36,18.1,19.13,20.28,21.73,22.69,23.63,25.74,25.82,26.31,32.15,34.26,$

$36.66,43.01,46.12,79.05\};$

The output of the above program code is as follows

$\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \text {Distribution} &{} \text {Parameters Estimates} &{} \text {-LL} &{} \text {AIC} &{} \text {BIC} \\ \text {LE} &{} \{\theta \rightarrow 1.5688,\lambda \rightarrow 0.109394\} &{} 412.049 &{} 828.099 &{} 833.803 \\ \text {PL} &{} \{\alpha \rightarrow 0.830204,\beta \rightarrow 0.294326\} &{} 413.354 &{} 830.708 &{} 836.412 \\ \text {L} &{} \{\theta \rightarrow 0.196045\} &{} 419.53 &{} 841.06 &{} 843.912 \\ \text {NGLD} &{} \{\alpha \rightarrow 1.17251,\theta \rightarrow 0.125193,\beta \rightarrow 1.17251\} &{} 413.368 &{} 832.736 &{} 841.292 \\ \end{array} \right)$