# A Compound Class of the Inverse Gamma and Power Series Distributions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

## 3. Estimation

#### 3.1. Quantile-Matching Estimation Method

#### 3.2. EM-Type Algorithm

- E-step: For $i=1,\dots ,n$, define ${\nu}_{ik}={\theta}^{(k-1)}\left(\right)open="("\; close=")">1-G\left(\right)open="("\; close=")">\frac{{\beta}^{(k-1)}}{{t}_{i}};{\alpha}^{(k-1)}$ and compute$$\begin{array}{c}\hfill {\tilde{M}}_{i}^{\left(k\right)}=\left(\right)open="\{"\; close>\begin{array}{c}{\displaystyle \frac{1+q{\nu}_{ik}}{1+{\nu}_{ik}},\mathrm{if}{M}_{i}\sim \mathrm{Binominal}}\hfill \\ {\displaystyle 1+{\nu}_{ik},\mathrm{if}{M}_{i}\sim \mathrm{Poisson}}\hfill \\ {\displaystyle \frac{1+{\nu}_{ik}}{1-{\nu}_{ik}},\mathrm{if}{M}_{i}\sim \mathrm{Geometric}}\hfill \\ {\displaystyle \frac{{\left(\right)}^{1}2}{{log}^{2}}-{\nu}_{ik}\left(\right)open="("\; close=")">2{\nu}_{ik}-1log\left(\right)open="("\; close=")">1-{\nu}_{ik}}\hfill & \left(\right)open="("\; close=")">1-{\nu}_{ik}log\left(\right)open="("\; close=")">1-{\nu}_{ik}\\ \left(\right)open="["\; close="]">\left(\right)open="("\; close=")">1-{\nu}_{ik}\end{array}log\left(\right)open="("\; close=")">1-{\nu}_{ik}-{\nu}_{ik}\end{array}$$
- M-step I: Update ${\theta}^{\left(k\right)}$ as the solution for the non-linear equation$$\frac{\theta {A}^{\prime}\left(\theta \right)}{A\left(\theta \right)}=\sum _{i=1}^{n}{M}_{i}^{\left(k\right)},$$
**CM-step II**: Given ${\beta}^{(k-1)}$, update ${\alpha}^{\left(k\right)}$ as$$\begin{array}{c}\hfill {\alpha}^{\left(k\right)}=arg\underset{\alpha}{max}\{n\left(\right)open="["\; close="]">\alpha log\left({\beta}^{(k-1)}\right)-log\mathrm{\Gamma}\left(\alpha \right)+\sum _{i=1}^{n}[\left(\right)open="("\; close=")">{\tilde{M}}_{i}^{\left(k\right)}-1\end{array}\hfill log\left(\right)open="("\; close=")">1-G\left(\right)open="("\; close=")">\frac{{\beta}^{(k-1)}}{{t}_{i}};\alpha & -(\alpha +1)log{t}_{i}\left]\right\}.$$**CM-step III**: Given ${\alpha}^{\left(k\right)}$, update ${\beta}^{\left(k\right)}$ as the solution for the non-linear equation$$\frac{n{\alpha}^{\left(k\right)}}{{\beta}^{\left(k\right)}}+\sum _{i=1}^{n}\left(\right)open="["\; close="]">\frac{1}{{t}_{i}}-\frac{{{\beta}^{\left(k\right)}}^{{\alpha}^{\left(k\right)}}\phantom{\rule{4pt}{0ex}}{t}_{i}^{-({\alpha}^{\left(k\right)}+1)}\phantom{\rule{4pt}{0ex}}{e}^{-\left(\right)open="("\; close=")">\frac{{\beta}^{\left(k\right)}}{t}}}{\left(\right)}$$- If some convergence condition is satisfied then stop iterating, otherwise move back to the E-step for another iteration.The standard errors of the estimates $\widehat{\psi}=(\widehat{\theta},\widehat{\alpha},\widehat{\beta})$ can be estimated using the method given by Louis [19]. Here, we use the observed information matrix instead of the Fisher’s information matrix and replace the missing values by the corresponding pseudo-values calculated in the last iteration of the ECM algorithm.

#### 3.3. Randomized Quantile Residuals

## 4. Simulation Study

## 5. Real Data Illustration

#### 5.1. Repair Times Data Set

#### 5.2. Gauge Lengths Data Set

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Adamidis, K.; Loukas, S. A Lifetime Distribution with Decreasing Failure Rate. Stat. Probab. Lett.
**1998**, 39, 35–42. [Google Scholar] [CrossRef] - Kus, C. A new lifetime distribution. Comput. Stat. Data Anal.
**2007**, 51, 4497–4509. [Google Scholar] [CrossRef] - Tahmasbi, R.; Rezaei, S. A two-parameter lifetime distribution with decreasing failure rate. Comput. Stat. Data Anal.
**2008**, 52, 3889–3901. [Google Scholar] [CrossRef] - Chahkandi, M.; Ganjali, M. On some lifetime distributions with decreasing failure rate. Comput. Stat. Data Anal.
**2009**, 53, 4433–4440. [Google Scholar] [CrossRef] - Morais, A.L.; Barreto-Souza, W. A compound class of Weibull and power series distribution. Comput. Stat. Data Anal.
**2011**, 55, 1410–1425. [Google Scholar] [CrossRef] - Mahmoudi, E.; Jafari, A.A. Generalized exponential power series distributions. Comput. Stat. Data Anal.
**2012**, 56, 4047–4066. [Google Scholar] [CrossRef] - Silva, R.B.; Bourguignon, M.; Dias, C.R.B.; Cordeiro, G.M. The compound family of extended Weibull power series distributions. Comput. Stat. Data Anal.
**2013**, 58, 352–367. [Google Scholar] [CrossRef] [Green Version] - Bagheri, S.F.; Samani, E.B.; Ganjali, M. The generalized modified Weibull power series distribution: Theory and applications. Comput. Stat. Data Anal.
**2016**, 94, 136–160. [Google Scholar] [CrossRef] - Warahena-Liyanage, G.; Pararai, M. The Lindley Power Series Class of Distributions: Model. Properties and Applications. J. Comput. Model.
**2015**, 5, 35–80. [Google Scholar] - Alizadeh, M.; Bagheri, S.F.; Bahrami-Samani, E.; Ghobadi, S.; Nadarajah, S. Exponentiated power Lindley power series class of distributions: Theory and applications. Commun.-Stat.-Simul. Comput.
**2018**, 47, 2499–2531. [Google Scholar] [CrossRef] - Elbatal, I.; Zayedm, M.; Rasekhi, M.; Butt, N.S. The Exponential Pareto Power Series Distribution: Theory and Applications. Pak. J. Stat. Oper. Res.
**2017**, 13, 603–615. [Google Scholar] [CrossRef] [Green Version] - Elbatal, I.; Altun, E.; Afify, A.Z.; Ozel, G. The Generalized Burr XII Power Series Distributions with Properties and Applications. Ann. Data Sci.
**2019**, 6, 571–597. [Google Scholar] [CrossRef] - Noack, A. On a class of discrete random variables. Ann. Math. Stat.
**1950**, 21, 127–132. [Google Scholar] [CrossRef] - Patil, G.P. Certain Properties of the Generalized Power Series Distribution. Ann. Math. Stat.
**1962**, 21, 179–182. [Google Scholar] [CrossRef] - Barakat, H.M.; Abdelkader, Y.H. Computing the moments of order statistics from nonidentical random variables. Stat. Methods Appl.
**2004**, 13, 15–26. [Google Scholar] [CrossRef] - Dempster, A.P.; Laird, N.M.; Rubim, D.B. Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. R. Stat. Soc. Ser. B
**1977**, 39, 1–38. [Google Scholar] - Gallardo, D.I.; Romeo, J.S.; Meyer, R. A simplified estimation procedure based on the EM algorithm for the power series cure rate model. Commun. Stat.-Simul. Comput.
**2017**, 46, 6342–6359. [Google Scholar] [CrossRef] - Meng, X.; Rubin, D. Maximum Likelihood Estimation via the ECM Algorithm: A General Framework. Biometrika
**1993**, 80, 267–278. [Google Scholar] [CrossRef] - Louis, T. Finding the observed information matrix when using the EM algorithm. J. R. Stat. Soc. Ser. B
**1982**, 44, 226–233. [Google Scholar] - Dunn, P.K.; Smyth, G.K. Randomized Quantile Residuals. J. Comput. Graph. Stat.
**1996**, 5, 236–244. [Google Scholar] - Von Alven, W.H. Reliability Engineering by ARINC; Prentice Hall: Upper Saddle River, NJ, USA, 1964. [Google Scholar]
- Akaike, H. A new look at the statistical model identification. IEEE Trans. Autom. Control
**1974**, 19, 716–723. [Google Scholar] [CrossRef] - Schwarz, G. Estimating the Dimension of a Model. Ann. Stat.
**1978**, 6, 461–464. [Google Scholar] [CrossRef] - Badar, M.G.; Priest, A.M. Statistical aspects of fiber and bundle strength in hybrid composites. In Progress in Science and Engineering Composites; Hayashi, T., Kawata, K., Umekawa, S., Eds.; ICCM-IV: Tokyo, Japan, 1982; pp. 1129–1136. [Google Scholar]
- Kundu, D.; Raqab, M.Z. Estimation of R = P(Y < X) for three-parameter Weibull distribution. Stat. Probab. Lett.
**2009**, 79, 1839–1846. [Google Scholar]

**Figure 1.**Density and hazard functions for the IGP, IGL, IGG and IGB distributions with different combinations for parameters.

**Figure 2.**(

**a**) Density function for IGG, WG and EP models and (

**b**) for the right tail in repair times data set.

**Figure 3.**QQ-plot of the randomized quantile residuals of IGG distribution for repair times data set.

**Figure 5.**QQ-plot of the randomized quantile residuals of IGG distribution for gauge lengths data set.

**Table 1.**Special cases of the PS$(\theta ,A(\theta \left)\right)$ distribution. For Binomial distribution q is considered known.

Distribution | Notation | ${\mathit{a}}_{\mathit{m}}$ | $\mathit{A}\left(\mathit{\theta}\right)$ | $\mathbf{\Theta}$ |
---|---|---|---|---|

Binomial | Bin($q,\theta )$ | $\left(\right)$ | ${(1+\theta )}^{q}-1$ | $(0,\infty )$ |

Poisson | Po($\theta $) | ${(m!)}^{-1}$ | ${e}^{\theta}-1$ | $(0,\infty )$ |

Geometric | Geo($\theta $) | 1 | $\theta {(1-\theta )}^{-1}$ | $(0,1)$ |

Logarithmic | Lo($\theta $) | ${\left(m\right)}^{-1}$ | $-log(1-\theta )$ | $(0,1)$ |

True Value | $\mathit{n}=50$ | $\mathit{n}=100$ | $\mathit{n}=200$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Model | $\mathit{\theta}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{bias}$ | $\mathit{se}$ | $\mathit{RMSE}$ | $\mathit{bias}$ | $\mathit{se}$ | $\mathit{RMSE}$ | $\mathit{bias}$ | $\mathit{se}$ | $\mathit{RMSE}$ | |

$IGP$ | 1.5 | 1.2 | 5 | $\widehat{\theta}$ | −0.193 | 1.999 | 1.562 | −0.036 | 1.537 | 1.496 | −0.004 | 1.375 | 1.369 |

$\widehat{\alpha}$ | 0.117 | 0.538 | 0.467 | 0.039 | 0.398 | 0.383 | 0.016 | 0.341 | 0.329 | ||||

$\widehat{\beta}$ | 0.139 | 1.218 | 1.182 | 0.084 | 0.833 | 0.806 | 0.063 | 0.579 | 0.562 | ||||

10 | $\widehat{\theta}$ | −0.182 | 2.003 | 1.631 | −0.101 | 1.574 | 1.511 | −0.054 | 1.417 | 1.380 | |||

$\widehat{\alpha}$ | 0.113 | 0.535 | 0.474 | 0.068 | 0.408 | 0.381 | 0.038 | 0.348 | 0.324 | ||||

$\widehat{\beta}$ | 0.255 | 2.625 | 2.440 | 0.070 | 1.704 | 1.598 | 0.059 | 1.152 | 1.097 | ||||

2 | 5 | $\widehat{\theta}$ | −0.161 | 2.032 | 1.532 | −0.111 | 1.609 | 1.500 | −0.004 | 1.440 | 1.413 | ||

$\widehat{\alpha}$ | 0.167 | 0.840 | 0.736 | 0.073 | 0.635 | 0.583 | 0.024 | 0.545 | 0.529 | ||||

$\widehat{\beta}$ | 0.148 | 1.306 | 1.162 | 0.078 | 0.835 | 0.772 | 0.067 | 0.656 | 0.624 | ||||

10 | $\widehat{\theta}$ | −0.123 | 2.038 | 1.793 | −0.064 | 1.812 | 1.692 | −0.027 | 1.522 | 1.499 | |||

$\widehat{\alpha}$ | 0.139 | 0.831 | 0.749 | 0.050 | 0.664 | 0.632 | 0.007 | 0.584 | 0.553 | ||||

$\widehat{\beta}$ | 0.204 | 2.492 | 2.380 | 0.142 | 1.690 | 1.637 | 0.126 | 1.309 | 1.262 | ||||

3 | 1.2 | 5 | $\widehat{\theta}$ | −0.636 | 2.930 | 2.564 | −0.346 | 2.683 | 2.396 | −0.057 | 2.372 | 2.201 | |

$\widehat{\alpha}$ | 0.349 | 0.800 | 0.697 | 0.219 | 0.629 | 0.599 | 0.117 | 0.559 | 0.531 | ||||

$\widehat{\beta}$ | 0.367 | 1.496 | 1.322 | 0.169 | 1.058 | 0.978 | 0.070 | 0.833 | 0.794 | ||||

10 | $\widehat{\theta}$ | −0.609 | 2.959 | 2.728 | −0.436 | 2.494 | 2.310 | −0.104 | 2.231 | 2.181 | |||

$\widehat{\alpha}$ | 0.340 | 0.885 | 0.705 | 0.233 | 0.631 | 0.577 | 0.114 | 0.511 | 0.503 | ||||

$\widehat{\beta}$ | 0.640 | 2.973 | 2.578 | 0.371 | 2.039 | 1.869 | 0.087 | 1.536 | 1.496 | ||||

2 | 5 | $\widehat{\theta}$ | −0.606 | 3.045 | 2.869 | −0.192 | 2.899 | 2.741 | −0.114 | 2.299 | 2.224 | ||

$\widehat{\alpha}$ | 0.504 | 1.364 | 1.103 | 0.300 | 1.131 | 0.954 | 0.178 | 0.846 | 0.789 | ||||

$\widehat{\beta}$ | 0.406 | 1.640 | 1.417 | 0.215 | 1.194 | 1.072 | 0.084 | 0.874 | 0.842 | ||||

10 | $\widehat{\theta}$ | −0.427 | 3.241 | 2.957 | −0.290 | 2.732 | 2.530 | −0.051 | 2.293 | 2.245 | |||

$\widehat{\alpha}$ | 0.433 | 1.345 | 1.114 | 0.310 | 1.025 | 0.926 | 0.169 | 0.873 | 0.820 | ||||

$\widehat{\beta}$ | 0.716 | 2.972 | 2.796 | 0.416 | 2.157 | 2.067 | 0.146 | 1.737 | 1.709 | ||||

$IGG$ | 0.2 | 1.2 | 5 | $\widehat{\theta}$ | 0.076 | 0.590 | 0.293 | 0.037 | 0.458 | 0.245 | 0.029 | 0.339 | 0.203 |

$\widehat{\alpha}$ | −0.026 | 0.404 | 0.326 | −0.021 | 0.285 | 0.222 | −0.020 | 0.202 | 0.164 | ||||

$\widehat{\beta}$ | 0.491 | 1.328 | 1.268 | 0.233 | 0.891 | 0.855 | 0.132 | 0.631 | 0.599 | ||||

10 | $\widehat{\theta}$ | 0.072 | 0.594 | 0.289 | 0.048 | 0.453 | 0.250 | 0.001 | 0.351 | 0.198 | |||

$\widehat{\alpha}$ | −0.044 | 0.405 | 0.309 | −0.029 | 0.285 | 0.228 | 0.000 | 0.202 | 0.161 | ||||

$\widehat{\beta}$ | 0.917 | 2.750 | 2.540 | 0.418 | 1.769 | 1.654 | 0.250 | 1.269 | 1.192 | ||||

2 | 5 | $\widehat{\theta}$ | 0.081 | 0.621 | 0.303 | 0.053 | 0.483 | 0.255 | 0.037 | 0.362 | 0.214 | ||

$\widehat{\alpha}$ | −0.038 | 0.659 | 0.535 | −0.032 | 0.459 | 0.357 | −0.021 | 0.324 | 0.258 | ||||

$\widehat{\beta}$ | 0.387 | 1.374 | 1.179 | 0.181 | 0.818 | 0.787 | 0.103 | 0.577 | 0.548 | ||||

10 | $\widehat{\theta}$ | 0.073 | 0.621 | 0.301 | 0.051 | 0.480 | 0.262 | 0.027 | 0.367 | 0.209 | |||

$\widehat{\alpha}$ | −0.022 | 0.652 | 0.515 | −0.018 | 0.460 | 0.380 | −0.014 | 0.325 | 0.254 | ||||

$\widehat{\beta}$ | 0.787 | 2.350 | 2.279 | 0.477 | 1.644 | 1.615 | 0.233 | 1.161 | 1.146 | ||||

0.85 | 1.2 | 5 | $\widehat{\theta}$ | −0.145 | 0.351 | 0.290 | −0.065 | 0.239 | 0.194 | −0.026 | 0.145 | 0.126 | |

$\widehat{\alpha}$ | 0.345 | 0.968 | 0.840 | 0.163 | 0.615 | 0.571 | 0.066 | 0.438 | 0.412 | ||||

$\widehat{\beta}$ | 0.265 | 1.374 | 1.170 | 0.107 | 0.789 | 0.722 | 0.059 | 0.499 | 0.492 | ||||

10 | $\widehat{\theta}$ | −0.150 | 0.336 | 0.299 | −0.060 | 0.204 | 0.189 | −0.032 | 0.128 | 0.117 | |||

$\widehat{\alpha}$ | 0.337 | 0.971 | 0.801 | 0.140 | 0.651 | 0.557 | 0.085 | 0.420 | 0.390 | ||||

$\widehat{\beta}$ | 0.504 | 2.448 | 2.234 | 0.181 | 1.536 | 1.412 | 0.094 | 1.053 | 1.041 | ||||

2 | 5 | $\widehat{\theta}$ | −0.126 | 0.303 | 0.283 | −0.079 | 0.234 | 0.217 | −0.033 | 0.147 | 0.132 | ||

$\widehat{\alpha}$ | 0.418 | 1.396 | 1.224 | 0.266 | 0.912 | 0.889 | 0.118 | 0.636 | 0.632 | ||||

$\widehat{\beta}$ | 0.267 | 1.324 | 1.159 | 0.119 | 0.866 | 0.774 | 0.058 | 0.533 | 0.520 | ||||

10 | $\widehat{\theta}$ | −0.131 | 0.304 | 0.278 | −0.073 | 0.238 | 0.213 | −0.031 | 0.135 | 0.129 | |||

$\widehat{\alpha}$ | 0.461 | 1.301 | 1.198 | 0.249 | 0.950 | 0.926 | 0.112 | 0.637 | 0.616 | ||||

$\widehat{\beta}$ | 0.583 | 2.450 | 2.286 | 0.267 | 1.618 | 1.564 | 0.097 | 1.065 | 1.042 |

**Table 3.**Maximum of the log-likelihood function ${\ell}_{max}$, AIC and BIC for EPS, WPS and IGPS models in the repair times data set.

Model | ${\mathit{\ell}}_{\mathit{max}}$ | AIC | BIC |
---|---|---|---|

EP | −102.8323 | 209.6645 | 213.3218 |

EL | −103.6670 | 211.3341 | 214.9914 |

EG | −103.2994 | 210.5988 | 214.2561 |

WP | −102.4637 | 210.9274 | 216.4133 |

WL | −103.7914 | 213.5828 | 219.0687 |

WG | −100.8561 | 207.7121 | 213.1981 |

IGP | −100.0756 | 206.1512 | 211.6371 |

IGL | −100.1348 | 206.2695 | 211.7555 |

IGG | −99.8685 | 205.7370 | 211.2229 |

**Table 4.**Estimates, standard errors (in brackets) and p-values associated with the AD, SF and SW statistics for IGG, WG and EP models in the repair times data set.

Parameter | IGG | WG | EP |
---|---|---|---|

$\widehat{\theta}$ | 0.6717 (0.3289) | 0.9667 (0.0540) | — |

$\widehat{\alpha}$ | 1.3924 (0.3041) | 1.4858 (0.2085) | 3.4288 (3.0519) |

$\widehat{\beta}$ | 0.9425 (0.4034) | 18.8997 (16.485) | 0.1080 (0.0910) |

p-value | |||

AD | 0.6088 | 0.2946 | 0.1126 |

SF | 0.7329 | 0.3989 | 0.0972 |

SW | 0.7336 | 0.3339 | 0.0823 |

**Table 5.**Maximum of the log-likelihood function ${\ell}_{max}$, AIC and BIC for EPS, WPS and IGPS models in the gauge lengths data set.

Model | ${\mathit{\ell}}_{\mathit{max}}$ | AIC | BIC |
---|---|---|---|

LL | −96.4058 | 196.8116 | 201.0979 |

LG | −59.4627 | 122.9254 | 127.2117 |

WP | −59.1711 | 124.3423 | 130.7717 |

WL | −61.2969 | 128.5939 | 135.0233 |

WG | −57.5006 | 121.0012 | 127.4306 |

IGP | −56.2875 | 118.5752 | 125.0046 |

IGL | −56.5613 | 119.1226 | 125.5520 |

IGG | −56.2871 | 118.5743 | 125.0037 |

**Table 6.**Estimates, standard errors (in brackets) and p-values associated with the AD, SF and SW statistics for IGG, WG and EP models in the gauge lengths data set.

Parameter | IGG | WG | LG |
---|---|---|---|

$\widehat{\theta}$ | 0.0102 (0.9237) | 0.9717 (0.0429) | 0.9997 (0.0003) |

$\widehat{\alpha}$ | 26.0899 (4.9419) | 8.3301 (1.0098) | 3.0636 (0.2973) |

$\widehat{\beta}$ | 76.6826 (13.8529) | 4.6071 (0.6742) | — |

p-value | |||

AD | 0.5279 | 0.2758 | 0.0954 |

SF | 0.8328 | 0.4633 | 0.1339 |

SW | 0.8874 | 0.4681 | 0.1096 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rivera, P.A.; Calderín-Ojeda, E.; Gallardo, D.I.; Gómez, H.W.
A Compound Class of the Inverse Gamma and Power Series Distributions. *Symmetry* **2021**, *13*, 1328.
https://doi.org/10.3390/sym13081328

**AMA Style**

Rivera PA, Calderín-Ojeda E, Gallardo DI, Gómez HW.
A Compound Class of the Inverse Gamma and Power Series Distributions. *Symmetry*. 2021; 13(8):1328.
https://doi.org/10.3390/sym13081328

**Chicago/Turabian Style**

Rivera, Pilar A., Enrique Calderín-Ojeda, Diego I. Gallardo, and Héctor W. Gómez.
2021. "A Compound Class of the Inverse Gamma and Power Series Distributions" *Symmetry* 13, no. 8: 1328.
https://doi.org/10.3390/sym13081328