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On the Beta-G Poisson Family

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Abstract

In this article, we propose and study a new family of distributions which is defined by using the genesis of the truncated Poisson distribution and the beta distribution. Some mathematical properties of the new family including moments, quantile and generating functions, mean deviations, order statistics and their moments, and reliability analysis are discussed. We also discuss the parameter estimation procedures and potential applications of such generalized family of distributions.

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Acknowledgements

The authors are grateful to the editor and anonymous reviewers for their constructive comments and valuable suggestions which certainly improved the presentation and quality of the paper.

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Correspondence to Gokarna R. Aryal.

Appendices

Appendix A

Elements of the observed information matrix \(J(\varvec{\varPhi })\) are given below:

$$\begin{aligned} U_{\lambda \lambda }= & {} -\frac{n}{\lambda ^2}+\frac{n\exp (-\lambda )}{[1-\exp (-\lambda )]^2}\\ U_{\lambda a}= & {} -\sum _{i=1}^n I^{a}_{G(x_i; \varvec{\varphi })}(a,b)\\ U_{\lambda b}= & {} -\sum _{i=1}^n I^{b}_{G(x_i;\varvec{\varphi })}(a,b)\\ U_{\lambda \varvec{\varphi }_r}= & {} -\sum _{i=1}^nI^{\varvec{\varphi }_r}_{G(x_i; \varvec{\varphi })}(a,b)\\ U_{aa}= & {} n\left[ \sum _{n=0}^\infty \frac{1}{(n+a+b)^2}-\sum _{i=0}^\infty \frac{1}{(n+a)^2}\right] -\lambda \sum _{i=1}^nI^{aa}_{G(x_i;\varvec{\varphi })}(a,b)\\ U_{ab}= & {} n\left[ \sum _{n=0}^\infty \frac{1}{(n+a+b)^2}\right] -\lambda \sum _{i=1}^nI^{ab}_{G(x_i;\varvec{\varphi })}(a,b)\\ U_{a\varvec{\varphi }_r}= & {} -\lambda \sum _{i=1}^nI^{a\varvec{\varphi }_r}_{G(x_i; \varvec{\varphi })}(a,b)+\sum _{i=1}^{n}\,\frac{G_{\varphi _r}^{\prime }\left( x_{i};\varvec{\varphi }\right) }{ {G}\left( x_{i};\varvec{\varphi }\right) }\\ U_{bb}= & {} n\left[ \sum _{n=0}^\infty \frac{1}{(n+a+b)^2}-\sum _{i=0}^\infty \frac{1}{(n+b)^2}\right] -\lambda \sum _{i=1}^nI^{bb}_{G(x_i;\varvec{\varphi })}(a,b)\\ U_{b\varvec{\varphi }_r}= & {} -\lambda \sum _{i=1}^nI^{b\varvec{\varphi }_r}_{G(x_i; \varvec{\varphi })}(a,b)-\sum _{i=1}^{n}\,\frac{G_{\varvec{\varphi }_r}^{\prime }\left( x_{i};\varvec{\varphi }\right) }{1- {G}\left( x_{i};\varvec{\varphi }\right) }\\ U_{\varvec{\varphi }_r\varvec{\varphi }_l}= & {} \sum _{i=1}^n\frac{g(x_i;\varvec{\varphi })g^{''}_{\varvec{\varphi }_r\varvec{\varphi }_l}(x_i;\varvec{\varphi })-g^{'}_{\varvec{\varphi }_r}\left( x_{i};\varvec{\varphi }\right) g^{'}_{\varvec{\varphi }_l}\left( x_{i};\varvec{\varphi }\right) }{[g(x_i;{\varvec{\varphi }})] ^2}-\lambda \sum _{i=1}^nI^{\varvec{\varphi }_r\varvec{\varphi }_l}_{G(x_i;\varvec{\varphi })}(a,b)\\&+\,(a-1)\sum _{i=1}^n\frac{G(x_i;\varvec{\varphi })G^{''}_{\varvec{\varphi }_r\varvec{\varphi }_l}(x_i;\varvec{\varphi })-G^{'}_{\varvec{\varphi }_r}\left( x_{i};\varvec{\varphi }\right) G^{'}_{\varvec{\varphi }_l}\left( x_{i};\varvec{\varphi }\right) }{[G(x_i;{\varvec{\varphi }})] ^2}\\&-\,(b-1)\sum _{i=1}^n\frac{[1-G(x_i;\varvec{\varphi })]G^{''}_{\varvec{\varphi }_r\varvec{\varphi }_l}(x_i;\varvec{\varphi })+G^{'}_{\varvec{\varphi }_r}\left( x_{i};\varvec{\varphi }\right) G^{'}_{\varvec{\varphi }_l}\left( x_{i};\varvec{\varphi }\right) }{[1-G(x_i;{\varvec{\varphi }})] ^2} \end{aligned}$$

Appendix B

Probability density functions of the distributions referenced in Sect. 7:

  • McW: \({f(x)=\frac{\beta \lambda \alpha ^{\beta }}{B\left( a/\lambda ,b\right) }\,x^{\beta -1}\,e^{-\left( \alpha x\right) ^{\beta }}\left[ 1-e^{-\left( \alpha x\right) ^{\beta }}\right] ^{a-1}\left\{ 1-\left( 1-e^{-\left( \alpha x\right) ^{\beta }}\right) ^{\lambda }\right\} ^{b-1} }\)

  • GTW: \(f(x)=\frac{\beta \alpha ^{\beta }x^{\beta -1}}{ \mathrm {e}^{(\alpha x)^{^{\beta }}}\left[ 1-\mathrm {e}^{-(\alpha x)^{^{\beta }}}\right] ^{1-a}}\left\{ a\left( 1+\lambda \right) -\lambda \left( a+b\right) \left[ 1-\mathrm {e}^{-(\alpha x)^{^{\beta }}}\right] ^{b}\right\} \)

  • TMW: \(f\left( x\right) =\,\left( \alpha +a\beta x^{\beta -1}\right) e^{-\left( \alpha x+ax^{\beta }\right) }\left\{ 1-\lambda +2\lambda e^{-\left( \alpha x+ax^{\beta }\right) }\right\} \)

  • TAW: \(f(x)=\)\(\frac{{\left( \alpha bx^{b-1}+a\beta x^{\beta -1}\right) }}{{e^{\left( \alpha x^{b}+ax^{\beta }\right) }}}{\left\{ 1-\lambda +2\lambda e^{-\left( \alpha x^{b}+ax^{\beta }\right) }\right\} }\)

  • TLL: \(f(x)=\beta \alpha ^{-\beta }x^{\beta -1}\left[ 1+\left( \frac{x}{\alpha }\right) ^\beta \right] ^{-2}\left[ 1-\lambda +2\lambda \{1+\left( \frac{x}{\alpha }\right) ^\beta \}^{-1}\right] \)

  • BLL: \(f(x)=\frac{\beta \alpha ^{-\beta }x^{\beta -1}}{B(a,b)}\left[ 1+\left( \frac{x}{\alpha }\right) ^\beta \right] ^{-2}\left[ 1-\{1+\left( \frac{x}{\alpha }\right) ^\beta \}^{-1}\right] ^{a-1}\left[ \{1{+}\left( \frac{x}{\alpha }\right) ^\beta \}^{-1}\right] ^{b-1}\)

  • KwLL :\(f(x)=\frac{ab\beta }{\alpha ^{a\beta }}x^{a\beta -1}\left[ 1+\left( \frac{x}{\alpha }\right) ^{\beta }\right] ^{-a-1} \left\{ 1-\left[ 1-\frac{1}{1+\left( \frac{x}{\alpha }\right) ^{\beta }}\right] ^{a} \right\} ^{b-1}\)

The parameters of the above densities are all positive real numbers except \( |\lambda |\le 1\) for GTW, TMW, TAW and TLL distributions.

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Aryal, G.R., Chhetri, S.B., Long, H. et al. On the Beta-G Poisson Family. Ann. Data. Sci. 6, 361–389 (2019). https://doi.org/10.1007/s40745-018-0176-x

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