# Generalising Exponential Distributions Using an Extended Marshall–Olkin Procedure

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## Abstract

**:**

## 1. Introduction

## 2. The Generalised Marshall–Olkin Exponential Distribution

**Definition**

**1.**

**Remark**

**1.**

**Remark**

**2.**

## 3. Some Properties of the GMO–Exponential Distribution

#### 3.1. Moments

`Mathematica`${}^{\copyright}$ software using the command

`HurwitzLerchPhi`.

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

- (i)
- ${\mu}_{n}=n!\delta \sum _{k=0}^{\infty}\frac{{\left(\right)}^{1}}{k}{\left(\right)}^{{\lambda}_{1}}n$
- (ii)
- In particular,$$E\left(X\right|{\lambda}_{1},{\lambda}_{2},\delta )=\frac{\delta}{{\lambda}_{2}}\mathsf{\Phi}(1-\delta ,1,\frac{{\lambda}_{1}}{{\lambda}_{2}}),$$$$Var\left(X\right|{\lambda}_{1},{\lambda}_{2},\delta )=\frac{{\delta}^{2}\mathsf{\Phi}{(1-\delta ,1,\frac{{\lambda}_{1}}{{\lambda}_{2}})}^{2}+2\delta \mathsf{\Phi}(1-\delta ,2,\frac{{\lambda}_{1}}{{\lambda}_{2}})}{{\lambda}_{2}^{2}}$$$${\kappa}_{3}=\frac{2{\delta}^{3}\mathsf{\Phi}{(1-\delta ,1,\frac{{\lambda}_{1}}{{\lambda}_{2}})}^{3}-6{\delta}^{2}\mathsf{\Phi}(1-\delta ,1,\frac{{\lambda}_{1}}{{\lambda}_{2}})\mathsf{\Phi}(1-\delta ,2,\frac{{\lambda}_{1}}{{\lambda}_{2}})+6\delta \mathsf{\Phi}(1-\delta ,3,\frac{{\lambda}_{1}}{{\lambda}_{2}})}{{\lambda}_{2}^{3}},$$
- (iii)
- For a fixed value of $\delta >0,$ the value of its kth moment decreases with ${\lambda}_{1}$ and with ${\lambda}_{2}$.

**Proof.**

#### 3.2. The Hazard Rate: Reliability Properties

**Proposition 2.**

- (i)
- For all ${\lambda}_{2}>0,$$$r\left(\right)open="("\; close=")">x|{\lambda}_{1},{\lambda}_{2},\delta iff\phantom{\rule{1.em}{0ex}}0\delta 1.$$
- (ii)
- For $x>0,$$$r\left(\right)open="("\; close=")">x|{\lambda}_{1},{\lambda}_{2},\delta iff{\lambda}_{1}\left(\right)open="["\; close="]">1-\left(\right)open="("\; close=")">1-\delta $$
- (iii)
- $r\left(\right)open="("\; close=")">x|{\lambda}_{1},{\lambda}_{2},\delta $ is a strictly decreasing function for $0<\delta <1,$ constant for $\delta =1$ and strictly increasing for $\delta >1.$

**Proof.**

#### 3.3. The Mode

- (Case a)
- If $0<\delta <1$, $N\left(\right)open="("\; close=")">t,\delta $ for any $t\ge 0$, and therefore the mode of distribution $\mathrm{GMOE}\left(\right)open="("\; close=")">{\lambda}_{1},{\lambda}_{2},\delta $ is reached at $M=0$
- (Case b)
- If $1<\delta <2,$$$N\left(\right)open="("\; close=")">t,\delta \left(\right)open="("\; close=")">2-\delta $$$$t<{s}_{1}=\frac{-\delta \left(\right)open="("\; close=")">\delta -1}{-}\left(\right)open="("\; close=")">\delta -1\left(\right)open="("\; close=")">2-\delta}0,$$$$t>{s}_{2}=\frac{-\delta \left(\right)open="("\; close=")">\delta -1}{+}\left(\right)open="("\; close=")">\delta -1\left(\right)open="("\; close=")">2-\delta}0;$$$$\frac{\delta \sqrt{\delta -1}-\delta \left(\right)open="("\; close=")">\delta -1}{}\left(\right)open="("\; close=")">\delta -1\left(\right)open="("\; close=")">2-\delta}{\lambda}_{1}{\lambda}_{2}.$$However, for $1<\delta <2$, the inequality$$\frac{\delta \sqrt{\delta -1}-\delta \left(\right)open="("\; close=")">\delta -1}{}\left(\right)open="("\; close=")">\delta -1\left(\right)open="("\; close=")">2-\delta}\frac{\delta}{\delta -1$$
- (Case c)
- If $\delta =2$, the condition reduces to $t>1$, or equivalently ${\lambda}_{1}<{\lambda}_{2}\le 2{\lambda}_{1}$.
- (Case d)
- Finally, if $\delta >2$, $N\left(\right)open="("\; close=")">t,\delta \left(\right)open="("\; close=")">\delta -2$ implies$$0<{s}_{1}=\frac{\delta \left(\right)open="("\; close=")">\delta -1}{-}\left(\right)open="("\; close=")">\delta -1\left(\right)open="("\; close=")">\delta -2$$

#### 3.4. Order Statistics

## 4. Estimation

`Mathematica`.

#### 4.1. Simulation Study

`OpenBUGS`is given in Appendix B. The summary statistics shown in Table 2 are based on $N=1000$ simulations with 50,000 iterations following a burn-in stage of 5000 iterations.

## 5. Numerical Illustrations

#### 5.1. Example 1

#### 5.1.1. Four (Three)-Parameter Gamma Distribution

#### 5.1.2. Three-Parameter Weibull Distribution

#### 5.2. Example 2

## 6. Extensions of the GMO Scheme

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

`OpenBUGS`code for the parameter estimation in GMOE distributions.

`model{`

`for (i in 1:n) {`

`dummy[i] <- 0`

`dummy[i] ~ dloglik(logLike[i]) likelihood is exp(logLike[i])`

`# log(likelihood)`

`logLike[i] <- -2*log(1-(1-delta)*exp(- lambda2*x[i])) +`

`log(lambda1*delta*exp(-lambda1*x[i])+`

`(lambda2-lambda1)*(1-delta)*delta*exp(-(lambda1+lambda2)*x[i]))`

`}`

`lambda1 ~ dunif(0,a)`

`lambda2 ~ dunif(0,b)`

`delta ~ dunif(0,c)`

`}`

## References

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**Figure 1.**Some probability density function (pdf) plots of generalised Marshall–Olkin exponential (GMOE) distributions for several values of $\delta $ parameter, $0<\delta \le 1$ (

**left panel**) and $\delta \ge 1$ (

**right panel**).

**Figure 2.**Some hazard rate plots of GMOE distributions for several values of $\delta $ parameter, $0<\delta \le 1$ (

**left panel**) and $\delta \ge 1$ (

**right panel**).

**Figure 3.**Ordinary histogram (

**left panel**) and empirical hazard function (

**right panel**) of data in Example 2.

n | Parameter | Mean | Bias | RMSE |
---|---|---|---|---|

25 | ${\lambda}_{1}$ | 0.56 | 0.19 | 0.63 |

${\lambda}_{2}$ | 0.98 | 6.76 | 61.96 | |

$\delta $ | 0.58 | 6908.21 | >10${}^{5}$ | |

50 | ${\lambda}_{1}$ | 0.54 | 0.07 | 0.37 |

${\lambda}_{2}$ | 1.01 | 2.18 | 9.74 | |

$\delta $ | 0.55 | 3814.02 | >10${}^{5}$ | |

100 | ${\lambda}_{1}$ | 0.51 | 0.016 | 0.19 |

${\lambda}_{2}$ | 1.003 | 1.11 | 5.25 | |

$\delta $ | 0.52 | 221.236 | >10${}^{3}$ |

n | Parameter | Mean | Bias | RMSE | StD |
---|---|---|---|---|---|

Scenario I : ${\lambda}_{1}=0.5,{\lambda}_{2}=1,\delta =0.5$ | |||||

25 | ${\lambda}_{1}$ | 0.60 | 0.10 | 0.62 | 0.13 |

${\lambda}_{2}$ | 0.99 | −0.01 | 0.99 | 0.15 | |

$\delta $ | 0.57 | 0.06 | 0.57 | 0.10 | |

50 | ${\lambda}_{1}$ | 0.58 | 0.08 | 0.59 | 0.13 |

${\lambda}_{2}$ | 1.02 | −0.01 | 1.03 | 0.17 | |

$\delta $ | 0.55 | 0.05 | 0.56 | 0.10 | |

100 | ${\lambda}_{1}$ | 0.54 | 0.07 | 0.55 | 0.10 |

${\lambda}_{2}$ | 1.007 | −0.01 | 1.08 | 0.18 | |

$\delta $ | 0.53 | 0.05 | 0.53 | 0.08 | |

Scenario II : ${\lambda}_{1}=1.5,{\lambda}_{2}=2.0,\delta =0.5$ | |||||

25 | ${\lambda}_{1}$ | 1.79 | 9.29 | 1.83 | 0.39 |

${\lambda}_{2}$ | 1.92 | −0.08 | 1.93 | 0.26 | |

$\delta $ | 0.58 | 0.07 | 0.58 | 0.10 | |

50 | ${\lambda}_{1}$ | 1.74 | 0.24 | 1.78 | 0.37 |

${\lambda}_{2}$ | 1.96 | −0.03 | 1.98 | 0.28 | |

$\delta $ | 0.57 | 0.07 | 0.58 | 0.10 | |

100 | ${\lambda}_{1}$ | 1.65 | 0.16 | 1.68 | 0.31 |

${\lambda}_{2}$ | 2.07 | 0.07 | 2.09 | 0.27 | |

$\delta $ | 0.55 | 0.05 | 0.56 | 0.09 |

**Table 3.**Dataset in Gupta and Kundu [4].

17.88 | 45.60 | 55.56 | 84.12 | 127.92 |

28.92 | 48.40 | 67.80 | 93.12 | 128.04 |

33.00 | 51.84 | 68.64 | 98.64 | 173.40 |

41.52 | 51.96 | 68.64 | 105.12 | |

42.12 | 54.12 | 68.88 | 105.84 |

Models | ||||
---|---|---|---|---|

Gamma | Weibull | 4–Parameters Gamma | GMOE | |

$(\widehat{\alpha},\widehat{\beta},\widehat{\gamma})$ | $(\widehat{\alpha},\widehat{\beta},\widehat{\mu})$ | $(\widehat{\alpha},\widehat{\beta},\widehat{\gamma},\widehat{\mu})$ | $({\widehat{\lambda}}_{1},{\widehat{\lambda}}_{2},\widehat{\delta})$ | |

Estimation | $(10.5669,1.3796,0.6032)$ | $(1.59,63.8723,14.8783)$ | $(0.7038,79.1267,1.8884,16.3264)$ | $(0.0359,0.0412,7.8156)$ |

Loglikelihood | $-112.97$ | $-112.85$ | $-112.847$ | $-113.42$ |

${\chi}^{2}$(d.f.) | 5.1739 (2) | 3.7826 (2) | 3.782 (3) | 3.00 (2) |

p-value | 0.6387 | 0.8044 | 0.804 | 0.223 |

BIC | 235.345 | 235.107 | 238.224 | 236.26 |

AIC | 231.938 | 231.70 | 283.682 | 232.85 |

Models | |||
---|---|---|---|

>Exponential | MOE | GMOE | |

$\widehat{\alpha}$ | $(\widehat{\lambda},\widehat{\delta})$ | $({\widehat{\lambda}}_{1},{\widehat{\lambda}}_{2},\widehat{\delta})$ | |

Estimation | $3.80153\times {10}^{-6}$ | $(1.96183\times {10}^{-7},0.003135)$ | $(3.12557\times {10}^{-6},4286.3,0.783562)$ |

Loglikelihood | $-29,022.7$ | $-\mathrm{26,638.8}$ | $-\mathrm{21,927.9}$ |

BIC | 58,053.1 | 53,292.9 | 43,878.8 |

AIC | 58,047.4 | 53,281.6 | 43,861.8 |

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**MDPI and ACS Style**

García, V.; Martel-Escobar, M.; Vázquez-Polo, F.J.
Generalising Exponential Distributions Using an Extended Marshall–Olkin Procedure. *Symmetry* **2020**, *12*, 464.
https://doi.org/10.3390/sym12030464

**AMA Style**

García V, Martel-Escobar M, Vázquez-Polo FJ.
Generalising Exponential Distributions Using an Extended Marshall–Olkin Procedure. *Symmetry*. 2020; 12(3):464.
https://doi.org/10.3390/sym12030464

**Chicago/Turabian Style**

García, Victoriano, María Martel-Escobar, and F.J. Vázquez-Polo.
2020. "Generalising Exponential Distributions Using an Extended Marshall–Olkin Procedure" *Symmetry* 12, no. 3: 464.
https://doi.org/10.3390/sym12030464