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On a class of hyper-Poisson and alternative hyper-Poisson distributions

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Abstract

Here we propose a class of hyper-Poisson and alternative hyper-Poisson distributions and study some of its important aspects by deriving expressions for its probability mass function, mean and variance and obtain conditions under which the distribution becomes under-dispersed or over-dispersed. Certain recurrence relations for probabilities, raw moments and factorial moments are also developed. Further, the estimation of the parameters of this class of hyper-Poisson distributions is attempted by various methods of estimation and shown that this new class of distribution gives better fit to certain real life data sets compared to the existing models.

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Acknowledgments

The authors would like to express their sincere gratitude to the Managing Editor and anonymous referees for their valuable comments on an earlier version of this paper which greatly improved the quality of the paper.

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Authors and Affiliations

  1. Department of Statistics, University of Kerala, Trivandrum, 695 581, India

    C. Satheesh Kumar

  2. Department of Statistics, M. S. M. College, Kayamkulam, Alappuzha, 690 502, India

    B. Unnikrishnan Nair

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  1. C. Satheesh Kumar
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  2. B. Unnikrishnan Nair
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Correspondence to C. Satheesh Kumar.

Appendix A

Appendix A

From Eq. 7 we have the first raw moment \(\mu _{1}\left (\lambda _{0}^{*}\right )= \mu _{1}(1,\lambda )\) is

$$\begin{array}{@{}rcl@{}} \mu_{1} (\lambda_{0}^{*} )&=&\lambda^{-1} \Lambda_{1} (\theta_{1} +2 \theta_{2} )\notag\\ &=&\frac{1}{\lambda}\frac{\phi(2;\lambda+1;\theta_{1}+\theta_{2}+\alpha)}{\phi(1;\lambda;\theta_{1}+\theta_{2}+\alpha)}(\theta_{1}+2\theta_{2}), \end{array} $$
(32)

by applying Eq. 9. Now by using Eq. 32, \(\mu _{1}\left (\lambda _{1}^{*}\right )= \mu _{1}(2, \lambda +1)\) can be written as

$$\begin{array}{@{}rcl@{}} \mu_{1} (\lambda_{1}^{*} )& =&\frac{2}{\lambda+1}\frac{\phi(3;\lambda+2;\theta_{1}+\theta_{2}+\alpha)}{\phi(2;\lambda+1;\theta_{1}+\theta_{2}+\alpha)}(\theta_{1}+2\theta_{2})\notag\\ &=&\frac{2}{\lambda+1}\frac{\Lambda_{2}}{\Lambda_{1}}\delta(1). \end{array} $$
(33)

Put r = 1 in Eq. 15 to get the following.

$$\begin{array}{@{}rcl@{}} \mu_{2} \left(\lambda_{0}^{*} \right)&=&\frac{1}{\lambda} \Lambda_{1}\left[(\theta_{1} +2 \theta_{2})\mu_{1}\left(\lambda_{1}^{*}\right)+\left(\theta_{1} +2^{2} \theta_{2}\right)\mu_{0}\left(\lambda_{1}^{*}\right) \right]\notag\\ &=&\frac{1}{\lambda}\frac{2}{\lambda+1}\Lambda_{2}\delta^{2}(1)+\frac{1}{\lambda}\Lambda_{1}\delta(2), \end{array} $$
(34)

in the light of Eq. 33 and

$$ \mu_{0}\left(\lambda_{1}^{*}\right)=1, $$
(35)

by definition. Equations 9 and 34 leads to the following.

$$\begin{array}{@{}rcl@{}} \mu_{2} \left(\lambda_{0}^{*} \right)&=&\frac{1}{\lambda}\frac{2}{(\lambda+1)}\frac{\phi(3;\lambda+2;\theta_{1}+\theta_{2}+\alpha)}{\phi(1;\lambda;\theta_{1}+\theta_{2}+\alpha)}\delta^{2} (1)\notag\\ &&+\frac{1}{\lambda}\frac{\phi(2;\lambda+1;\theta_{1}+\theta_{2}+\alpha)}{\phi(1;\lambda;\theta_{1}+\theta_{2}+\alpha)} \delta (2) \end{array} $$
(36)

By using Eq. 36, \( \mu _{2} \left (\lambda _{1}^{*} \right )=\mu _{2} (2, \lambda +1 ) \) can be written as

$$\begin{array}{@{}rcl@{}} \mu_{2} \left(\lambda_{1}^{*} \right)&=&\frac{2}{\lambda+1} \frac{3}{\lambda+2}\frac{\phi(4;\lambda+3;\theta_{1}+\theta_{2}+\alpha)}{\phi(2;\lambda+1;\theta_{1}+\theta_{2}+\alpha)}\delta^{2}(1)\\ && + \frac{2}{\lambda+1} \frac{\phi(3;\lambda+2;\theta_{1}+\theta_{2}+\alpha)}{\phi(2;\lambda+1;\theta_{1}+\theta_{2}+\alpha)}\delta (2)\notag\\ &=& \frac{2}{\lambda+1} \frac{3}{\lambda+2}\frac{\Lambda_{3}}{\Lambda_{1}}\delta^{2}(1) +\frac{2}{\lambda+1} \frac{\Lambda_{2}}{\Lambda_{1}}\delta (2) \end{array} $$
(37)

Put r = 2 in Eq. 15 to obtain

$$\begin{array}{@{}rcl@{}} \mu_{3} \left(\lambda_{0}^{*} \right) & =&\frac{1}{\lambda} \Lambda_{1}\left[(\theta_{1} +2 \theta_{2})\mu_{2}\left(\lambda_{1}^{*}\right)+2\left(\theta_{1} +2^{2} \theta_{2}\right)\mu_{1}\left(\lambda_{1}^{*}\right)\right.\\ &&\left.+ \left(\theta_{1} +2^{3} \theta_{2}\right)\mu_{0}\left(\lambda_{1}^{*}\right)\right]. \end{array} $$
(38)

On substituting (33), (35) and (37) in (38), we get

$$\begin{array}{@{}rcl@{}} \mu_{3} \left(\lambda_{0}^{*} \right)&=&\frac{1}{\lambda}\frac{2}{\lambda+1}\frac{3}{\lambda+2}\Lambda_{3}\delta^{3}(1)+3\frac{1}{\lambda}\frac{2}{\lambda+1}\Lambda_{2}\delta(1)\delta(2)+\frac{1}{\lambda}\Lambda_{1}\delta(3)\\ &=&\frac{1}{\lambda}\frac{2}{\lambda+1}\frac{3}{\lambda+2}\frac{\phi(4;\lambda+3;\theta_{1}+\theta_{2}+\alpha)}{\phi(1;\lambda;\theta_{1}+\theta_{2}+\alpha)}\delta^{3}(1)\\ &&+3\frac{1}{\lambda}\frac{2}{\lambda+1}\frac{\phi(3;\lambda;\theta_{1}+\theta_{2}+\alpha)}{\phi(1;\lambda;\theta_{1}+\theta_{2}+\alpha)}\delta(1)\delta(2)\notag\\ &&+\frac{1}{\lambda}\frac{\phi(2;\lambda+1;\theta_{1}+\theta_{2}+\alpha)}{\phi(1;\lambda;\theta_{1}+\theta_{2}+\alpha)}\delta(3) \end{array} $$
(39)

in the light of Eq. 9. Next, by using (39) \( \mu _{3} \left (\lambda _{1}^{*}\right )=\mu _{3} (2, \lambda +1)\) can be written as

$$\begin{array}{@{}rcl@{}} \mu_{3} \left(\lambda_{1}^{*} \right)&=&\frac{2}{\lambda+1} \frac{3}{\lambda+2}\frac{4}{\lambda+3}\frac{\phi(5;\lambda+4;\theta_{1}+\theta_{2}+\alpha)}{\phi(2;\lambda+1;\theta_{1}+\theta_{2}+\alpha)}\delta^{3}(1)\\ &&+3 \frac{2}{\lambda+1}\frac{3}{\lambda+2} \frac{\phi(4;\lambda+3;\theta_{1}+\theta_{2}+\alpha)}{\phi(2;\lambda+1;\theta_{1}+\theta_{2}+\alpha)}\delta(2)\\ &&\times\delta(1)\\ &&+\frac{2}{\lambda+1}\frac{\phi(3;\lambda+2;\theta_{1}+\theta_{2}+\alpha)}{\phi(2;\lambda+1;\theta_{1}+\theta_{2}+\alpha)}\delta(3)\\ &=&\frac{2}{\lambda+1} \frac{3}{\lambda+2}\frac{4}{\lambda+3}\frac{\Lambda_{4}}{\Lambda_{1}}\delta^{3}(1)+ 3\frac{2}{\lambda+1}\frac{3}{\lambda+2}\frac{\Lambda_{3}}{\Lambda_{1}}\delta(2)\delta(1)\notag\\ &&+\frac{2}{\lambda+1}\frac{\Lambda_{2}}{\Lambda_{1}}\delta(3) \end{array} $$
(40)

Put r = 3 in Eq. 15 to get

$$\begin{array}{@{}rcl@{}} \mu_{4} \left(\lambda_{0}^{*} \right) &=&\frac{1}{\lambda} \Lambda_{1}\left[(\theta_{1} +2 \theta_{2})\mu_{3}\left(\lambda_{1}^{*}\right)+3\left(\theta_{1} +2^{2} \theta_{2}\right)\mu_{2}\left(\lambda_{1}^{*}\right)\right.\\ &&\left.+ 3\left(\theta_{1} +2^{3} \theta_{2}\right)\mu_{1}\left(\lambda_{1}^{*}\right)+\left(\theta_{1} +2^{4} \theta_{2}\right)\mu_{0}\left(\lambda_{1}^{*}\right)\right] \end{array} $$
(41)

On substituting (33),(35),(37) and (40) in Eq. 41

$$\begin{array}{@{}rcl@{}} \mu_{4} \left(\lambda_{0}^{*} \right)&=&{\frac{24}{(\lambda)_{4}} \delta(1)^{4} \Lambda_{4} +\frac{6}{(\lambda)_{3}} \delta^{2}(1) \delta(2) \Lambda_{3} }{+\frac{8}{(\lambda)_{2}} \Lambda_{2} \left[\delta(3) \delta(1) +3\delta^{2}(2) \right]}\\ &&+\frac{1}{\lambda } \delta(4) \Lambda_{1} \end{array} $$
(42)

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Kumar, C.S., Nair, B.U. On a class of hyper-Poisson and alternative hyper-Poisson distributions. OPSEARCH 52, 86–100 (2015). https://doi.org/10.1007/s12597-013-0169-7

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