# Discrete Single-Factor Extension of the Exponential Distribution: Features and Modeling

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## Abstract

**:**

## 1. Introduction

## 2. The Structure of the DMEx Model

#### 2.1. Moments of a Statistical Distribution

#### 2.2. Dispersion and Variation Measures

#### 2.3. L-Moment Statistics

## 3. Different Estimation Techniques

#### 3.1. Maximum Likelihood Estimation (MLE)

#### 3.2. Moment Estimation (ME)

#### 3.3. Proportion Estimation (PE)

## 4. Comparing Different Estimators (CDEs): A Simulation Study

- 1.
- Generate $N=\mathrm{10,000}$ samples of various sizes “${n}_{i};i=1,2,3,4$” from the DMEx model as follows
- Scheme I: $\beta =0.15$|${n}_{1}=25,{n}_{2}=50,{n}_{3}=100,{n}_{4}=250,{n}_{5}=400,{n}_{6}=600.$
- Scheme II: $\beta =0.35$|${n}_{1}=25,{n}_{2}=50,{n}_{3}=100,{n}_{4}=250,{n}_{5}=400,{n}_{6}=600.$
- Scheme III: $\beta =0.85$|${n}_{1}=25,{n}_{2}=50,{n}_{3}=100,{n}_{4}=250,{n}_{5}=400,{n}_{6}=600.$

- 2.
- Compute the MLE, ME, and PE for the 10,000 samples, say ${\widehat{\beta}}_{k}$ for $k=1,2,\dots ,$ 10,000.
- 3.
- Calculate the bias “BS”, mean squared errors (MSE), and mean relative errors (MRE) for N = 10,000 samples as$$\left|\mathrm{BS}\left(\beta \right)\right|=\frac{1}{N}\sum _{k=1}^{N}\left|\widehat{{\beta}_{k}}-{\beta}_{k}\right|,\mathrm{MSE}\left(\beta \right)=\frac{1}{N}\sum _{k=1}^{N}{(\widehat{{\beta}_{k}}-{\beta}_{k})}^{2},\mathrm{MRE}\left(\beta \right)=\frac{1}{N}\sum _{k=1}^{N}\frac{\left|\widehat{{\beta}_{k}}-{\beta}_{k}\right|}{{\beta}_{k}}.$$
- 4.

## 5. Data Modeling: Competitive Models and Statistical Criteria

#### 5.1. Data Set I: Electronic Components

#### 5.2. Data Set II: Leukemia Remission

#### 5.3. Data Set III: Coronavirus in Punjab

## 6. Concluding Remarks and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bhatti, F.A.; Hamedani, G.G.; Korkmaz, M.Ç.; Sheng, W.; Ali, A. On the Burr XII-moment exponential distribution. PLoS ONE
**2021**, 16, e0246935. [Google Scholar] [CrossRef] [PubMed] - Iqbal, Z.; Hasnain, S.A.; Salman, M.; Ahmad, M.; Hamedani, G. Generalized exponentiated moment exponential distribution. Pak. J. Stat.
**2014**, 30, 537–554. [Google Scholar] - Ahsan-ul-Haq, M. On Poisson moment exponential distribution with applications. Ann. Data Sci.
**2022**, 1–22. [Google Scholar] [CrossRef] - Abbas, S.; Jahngeer, A.; Shahbaz, S.H.; Afify, A.Z. Topp-Leone moment exponential distribution: Properties and applications. 2020. J. Natl. Sci. Found. Sri Lanka
**2020**, 48, 265–274. [Google Scholar] [CrossRef] - Akhter, Z.; MirMostafaee, S.M.T.K.; Ormoz, E. On the order statistics of exponentiated moment exponential distribution and associated inference. J. Stat. Comput. Simul.
**2022**, 92, 1322–1346. [Google Scholar] [CrossRef] - Hashmi, S.; Haq, M.A.U.; Usman, R.M.; Ozel, G. The Weibull-moment exponential distribution: Properties, characterizations and applications. J. Reliab. Stat. Stud.
**2019**, 12, 1–22. [Google Scholar] - Kumar, D.; Dey, T.; Dey, S. Statistical inference of exponentiated moment exponential distribution based on lower record values. Commun. Math. Stat.
**2017**, 5, 231–260. [Google Scholar] [CrossRef] - Iriarte, Y.A.; Astorga, J.M.; Venegas, O.; Gómez, H.W. Slashed moment exponential distribution. J. Stat. Appl.
**2017**, 16, 354–365. [Google Scholar] [CrossRef] [Green Version] - Ibrahim, M.; Ali, M.M.; Yousof, H.M. The discrete analogue of the Weibull-G family: Properties, different applications, Bayesian and non-Bayesian estimation methods. Ann. Data Sci.
**2021**, 1–38. [Google Scholar] [CrossRef] - Eliwa, M.S.; El-Morshedy, M.; Yousof, H.M. A discrete exponential generalized-G family of distributions: Properties with Bayesian and non-Bayesian estimators to model medical, engineering and agriculture data. Mathematics
**2022**, 10, 3348. [Google Scholar] [CrossRef] - Yousof, H.M.; Chesneau, C.; Hamedani, G.; Ibrahim, M. A new discrete distribution: Properties, characterizations, modeling real count data, Bayesian and non-Bayesian estimations. Statistica
**2021**, 81, 135–162. [Google Scholar] [CrossRef] - Altun, E.; El-Morshedy, M.; Eliwa, M.S. A study on discrete Bilal distribution with properties and applications on integervalued autoregressive process. Revstat-Stat. J.
**2022**, 20, 501–528. [Google Scholar] - El-Morshedy, M.; Altun, E.; Eliwa, M.S. A new statistical approach to model the counts of novel coronavirus cases. Math. Sci.
**2022**, 16, 37–50. [Google Scholar] [CrossRef] [PubMed] - Almetwally, E.M.; Ibrahim, G.M. Discrete alpha power inverse Lomax distribution with application of COVID-19 data. J. Appl. Math.
**2020**, 9, 11–22. [Google Scholar] - Tyagi, A.; Choudhary, N.; Singh, B. A new discrete distribution: Theory and applications to discrete failure lifetime and count data. J. Appl. Probab. Stat.
**2020**, 15, 117–143. [Google Scholar] - Bakouch, H.S.; Jazi, M.A.; Nadarajah, S. A new discrete distribution. Statistics
**2014**, 48, 200–240. [Google Scholar] [CrossRef] - Johnston, G. Statistical Models and Methods for Lifetime Data; Wiley: New York, NY, USA, 2003. [Google Scholar]
- Damien, P.; Walker, S. A Bayesian non-parametric comparison of two treatments. Scand. J. Stat.
**2002**, 29, 51–56. [Google Scholar] [CrossRef]

$\begin{array}{c}\mathit{\beta}\u27f6\\ \mathrm{Measure}\downarrow \end{array}$ | $0.1$ | $0.2$ | $0.3$ | $0.4$ | $0.5$ | $0.6$ | $0.7$ | $0.8$ | $0.9$ |
---|---|---|---|---|---|---|---|---|---|

$E\left(X\right)$ | $0.3954$ | $0.7529$ | $1.1657$ | $1.6848$ | $2.3863$ | $3.4156$ | $5.1075$ | $8.4629$ | $18.4824$ |

$Var\left(X\right)$ | $0.3901$ | $0.8139$ | $1.4378$ | $2.4502$ | $4.2371$ | $7.7429$ | $15.802$ | $40.249$ | $180.2500$ |

$Sk\left(X\right)$ | $1.6228$ | $1.4038$ | $1.3664$ | $1.3697$ | $1.3821$ | $1.3943$ | $1.4037$ | $1.4099$ | $1.4132$ |

$Ku\left(X\right)$ | $9.8387$ | $14.3813$ | $19.6580$ | $25.0539$ | $30.3715$ | $35.5222$ | $40.4667$ | $40.4667$ | $49.7002$ |

$\begin{array}{c}\mathit{\beta}\u27f6\\ \mathrm{Measure}\downarrow \end{array}$ | $0.1$ | $0.2$ | $0.3$ | $0.4$ | $0.5$ | $0.6$ | $0.7$ | $0.8$ | $0.9$ |
---|---|---|---|---|---|---|---|---|---|

$D\left(X\right)$ | $0.9866$ | $1.0810$ | $1.2335$ | $1.4543$ | $1.7756$ | $2.2669$ | $3.0939$ | $4.7559$ | $9.7525$ |

$C\left(X\right)$ | $1.5797$ | $1.1982$ | $1.0287$ | $0.9291$ | $0.8626$ | $0.8147$ | $0.7783$ | $0.7497$ | $0.7264$ |

n | Criteria | MLE | ME | PE |
---|---|---|---|---|

25 | $\left|\mathrm{BS}\right|$ | $0.71072536$ | $0.68736626$ | $1.01333569$ |

MSE | $0.50610243$ | $0.48899665$ | $0.62541235$ | |

MRE | $0.47496366$ | $0.46614520$ | $0.67532023$ | |

50 | $\left|\mathrm{BS}\right|$ | $0.39401774$ | $0.38114550$ | $0.59444182$ |

MSE | $0.10203639$ | $0.09469954$ | $0.17296732$ | |

MRE | $0.26463284$ | $0.25413209$ | $0.39819026$ | |

100 | $\left|\mathrm{BS}\right|$ | $0.27723665$ | $0.25923302$ | $0.39711141$ |

MSE | $0.07412203$ | $0.06774263$ | $0.15838302$ | |

MRE | $0.18212052$ | $0.17334963$ | $0.26733503$ | |

250 | $\left|\mathrm{BS}\right|$ | $0.18533699$ | $0.18241249$ | $0.29293434$ |

MSE | $0.03441458$ | $0.03211288$ | $0.08771857$ | |

MRE | $0.12229566$ | $0.12014121$ | $0.1964149$ | |

400 | $\left|\mathrm{BS}\right|$ | $0.13130521$ | $0.11796928$ | $0.19905126$ |

MSE | $0.01796344$ | $0.01441778$ | $0.03983039$ | |

MRE | $0.08722563$ | $0.07803269$ | $0.1327154$ | |

600 | $\left|\mathrm{BS}\right|$ | $0.04210295$ | $0.03471560$ | $0.09222016$ |

MSE | $0.00429679$ | $0.00413072$ | $0.01223098$ | |

MRE | $0.00831602$ | $0.00726341$ | $0.02431204$ |

n | Criteria | MLE | ME | PE |
---|---|---|---|---|

25 | $\left|\mathrm{BS}\right|$ | $0.46033471$ | $0.45230219$ | $0.49111028$ |

MSE | $0.21236964$ | $0.20903288$ | $0.24800326$ | |

MRE | $0.93014126$ | $0.91314177$ | $0.99711412$ | |

50 | $\left|\mathrm{BS}\right|$ | $0.31830325$ | $0.31130925$ | $0.41790345$ |

MSE | $0.10344125$ | $0.09841329$ | $0.17471516$ | |

MRE | $0.63899659$ | $0.62332954$ | $0.82130958$ | |

100 | $\left|\mathrm{BS}\right|$ | $0.23141239$ | $0.22213412$ | $0.31815142$ |

MSE | $0.05210236$ | $0.04930287$ | $0.10274589$ | |

MRE | $0.46266369$ | $0.44310965$ | $0.63937195$ | |

250 | $\left|\mathrm{BS}\right|$ | $0.16141257$ | $0.15810236$ | $0.24730864$ |

MSE | $0.02695256$ | $0.02596985$ | $0.05886243$ | |

MRE | $0.32242856$ | $0.31519732$ | $0.48810236$ | |

400 | $\left|\mathrm{BS}\right|$ | $0.11463142$ | $0.09866367$ | $0.17199896$ |

MSE | $0.01386537$ | $0.00980015$ | $0.02914120$ | |

MRE | $0.22720103$ | $0.19710414$ | $0.33409875$ | |

600 | $\left|\mathrm{BS}\right|$ | $0.08795636$ | $0.07296985$ | $0.12110286$ |

MSE | $0.00877157$ | $0.00627420$ | $0.00923698$ | |

MRE | $0.11209537$ | $0.09830987$ | $0.21008025$ |

n | Criteria | MLE | ME | PE |
---|---|---|---|---|

25 | $\left|\mathrm{BS}\right|$ | $0.46296336$ | $0.48641129$ | $0.49813368$ |

MSE | $0.21374125$ | $0.23632653$ | $0.24841515$ | |

MRE | $0.92396326$ | $0.97291764$ | $0.99433695$ | |

50 | $\left|\mathrm{BS}\right|$ | $0.31103258$ | $0.37031526$ | $0.42356964$ |

MSE | $0.09774623$ | $0.13274859$ | $0.17233696$ | |

MRE | $0.62174120$ | $0.74533626$ | $0.81241852$ | |

100 | $\left|\mathrm{BS}\right|$ | $0.23163949$ | $0.26933026$ | $0.31233635$ |

MSE | $0.05208856$ | $0.07341259$ | $0.09810221$ | |

MRE | $0.46241203$ | $0.53233636$ | $0.61233982$ | |

250 | $\left|\mathrm{BS}\right|$ | $0.16196336$ | $0.19810775$ | $0.22185236$ |

MSE | $0.02641205$ | $0.03901486$ | $0.04941212$ | |

MRE | $0.32163955$ | $0.39910213$ | $0.44330352$ | |

400 | $\left|\mathrm{BS}\right|$ | $0.11300125$ | $0.13433636$ | $0.16241252$ |

MSE | $0.01322585$ | $0.01841021$ | $0.02744125$ | |

MRE | $0.23474694$ | $0.27117655$ | $0.33139625$ | |

600 | $\left|\mathrm{BS}\right|$ | $0.03332357$ | $0.09811494$ | $0.12141254$ |

MSE | $0.00894112$ | $0.00903661$ | $0.01230225$ | |

MRE | $0.10541453$ | $0.12322396$ | $0.18338552$ |

Model | $\mathit{\beta}$ | $\mathit{\alpha}$ | ||||
---|---|---|---|---|---|---|

MLE | SE | CI | MLE | SE | CI | |

DMEx | $0.931$ | $0.012$ | $[0.910,0.956]$ | − | − | − |

DR | $0.999$ | $2.6\times {10}^{-4}$ | $[0.998,0.999]$ | − | − | − |

DIR | $1.8\times {10}^{-7}$ | $0.055$ | $[0,0.107]$ | − | − | − |

DBH | $0.999$ | $0.008$ | $[0.984,1.014]$ | − | − | − |

DPa | $0.720$ | $0.061$ | $[0.600,0.839]$ | − | − | − |

Poi | $27.533$ | $1.355$ | $[24.878,30.189]$ | − | − | − |

DINH | $0.578$ | $0.193$ | $[0.199,0.957]$ | $29.072$ | $20.384$ | $[0,69.024]$ |

DB-XII | $0.975$ | $0.051$ | $[0.874,1]$ | $13.367$ | $27.785$ | $[0,67.824]$ |

Statistic | DMEx | DR | DIR | DBH | DPa | Poi | DINH | DB-XII |
---|---|---|---|---|---|---|---|---|

$-L$ | $64.7898$ | $66.394$ | $89.096$ | $91.368$ | $77.402$ | $151.206$ | $67.879$ | $75.724$ |

AIC | $131.5796$ | $134.788$ | $180.192$ | $184.737$ | $156.805$ | $304.413$ | $139.758$ | $155.448$ |

CAIC | $131.8873$ | $135.096$ | $180.499$ | $185.045$ | $157.112$ | $304.721$ | $140.758$ | $156.448$ |

BIC | $132.2877$ | $135.496$ | $180.899$ | $185.445$ | $157.513$ | $305.121$ | $141.174$ | $156.864$ |

HQIC | $131.5721$ | $134.781$ | $180.184$ | $184.729$ | $156.797$ | $304.405$ | $139.743$ | $155.433$ |

KS | $0.1144$ | $0.216$ | $0.698$ | $0.791$ | $0.405$ | $0.381$ | $0.207$ | $0.388$ |

p-value | $0.9766$ | $0.433$ | <$0.0001$ | <$0.0001$ | $0.009$ | $0.025$ | $0.481$ | $0.015$ |

Technique | $\mathit{\beta}$ | KS | p-Value |
---|---|---|---|

ME | $0.931$ | $0.114$ | $0.989$ |

PE | $0.811$ | $0.588$ | $0.627\times {10}^{-4}$ |

Approach | $\mathit{E}\left(\mathit{X}\right)$ | $\mathit{Var}\left(\mathit{X}\right)$ | $\mathit{D}\left(\mathit{X}\right)$ | $\mathit{Sk}\left(\mathit{X}\right)$ | $\mathit{Ku}\left(\mathit{X}\right)$ |
---|---|---|---|---|---|

MLE | $27.536$ | $393.082$ | $14.275$ | $1.414$ | $5.999$ |

PE | $9.0508$ | $45.6912$ | $5.048$ | $1.410$ | $5.989$ |

ME | $27.533$ | $393.017$ | $14.274$ | $1.414$ | $5.999$ |

Model | $\mathit{\beta}$ | $\mathit{\alpha}$ | ||||
---|---|---|---|---|---|---|

MLE | SE | CI | MLE | SE | CI | |

DMEx | $0.905$ | $0.014$ | $[0.877,0.933]$ | − | − | − |

DR | $0.998$ | $0.0004$ | $[0.998,0.999]$ | − | − | − |

DIR | $7.82\times {10}^{-7}$ | − | − | − | − | − |

DBH | $0.998$ | $0.009$ | $[0.981,1.017]$ | − | − | − |

DPa | $0.696$ | $0.056$ | $[0.585,0.806]$ | − | − | − |

Poi | $19.550$ | $0.989$ | $[17.612,21.493]$ | − | − | − |

DINH | $0.737$ | $0.268$ | $[0.212,1.262]$ | $14.798$ | $9.997$ | $[0,34.392]$ |

DB-XII | $0.998$ | $0.004$ | $[0.99,1]$ | $182.367$ | $94.801$ | $[0,277.7.001]$ |

Statistic | DMEx | DR | DIR | DBH | DPa | Poi | DINH | DB-XII |
---|---|---|---|---|---|---|---|---|

$-L$ | $79.071$ | $81.175$ | $101.987$ | $110.283$ | $95.448$ | $152.718$ | $82.818$ | $92.602$ |

AIC | $160.141$ | $164.351$ | $205.975$ | $222.565$ | $192.896$ | $307.436$ | $169.635$ | $189.203$ |

CAIC | $160.364$ | $164.572$ | $206.197$ | $222.787$ | $193.118$ | $307.658$ | $170.341$ | $189.909$ |

BIC | $161.137$ | $165.346$ | $206.973$ | $223.561$ | $193.892$ | $308.432$ | $171.627$ | $191.195$ |

HQIC | $160.336$ | $164.544$ | $206.169$ | $222.759$ | $193.090$ | $307.630$ | $170.024$ | $189.592$ |

KS | $0.109$ | $0.199$ | − | $0.751$ | $0.392$ | $0.352$ | $0.189$ | $0.369$ |

p-value | $0.970$ | $0.401$ | − | <0.001 | $0.004$ | $0.014$ | $0.467$ | $0.008$ |

Technique | $\mathit{\beta}$ | KS | p-Value |
---|---|---|---|

ME | $0.905$ | $0.100$ | $0.841$ |

PE | $0.837$ | $0.350$ | $0.0147$ |

Approach | $\mathit{E}\left(\mathit{X}\right)$ | $\mathit{Var}\left(\mathit{X}\right)$ | $\mathit{D}\left(\mathit{X}\right)$ | $\mathit{Sk}\left(\mathit{X}\right)$ | $\mathit{Ku}\left(\mathit{X}\right)$ |
---|---|---|---|---|---|

MLE | $19.551$ | $201.109$ | $10.286$ | $1.413$ | $5.998$ |

PE | $10.756$ | $63.433$ | $5.897$ | $1.411$ | $5.992$ |

ME | $19.550$ | $201.084$ | $10.286$ | $1.413$ | $5.998$ |

Model | $\mathit{\beta}$ | $\mathit{\alpha}$ | ||||
---|---|---|---|---|---|---|

MLE | SE | CI | MLE | SE | CI | |

DMEx | $0.9451$ | $0.006$ | $[0.93,0.957]$ | − | − | − |

DR | $0.9996$ | $0.00008$ | $[0.9994,0.999]$ | − | − | − |

DIR | $1.634\times {10}^{-10}$ | − | − | − | − | − |

DBH | $0.999$ | $0.004$ | $[0.993,1.006]$ | − | − | − |

DPa | $0.729$ | $0.037$ | $[0.658,0.803]$ | − | − | − |

Poi | $34.921$ | $0.959$ | $[33.04,36.8]$ | − | − | − |

DINH | $0.615$ | $0.144$ | $[0.333,0.896]$ | $29.319$ | $13.995$ | $[1.889,56.748]$ |

DB-XII | $0.996$ | $0.004$ | $[0.989,1.003]$ | $79.588$ | $82.339$ | $[0,215.023]$ |

Statistic | DMEx | DR | DIR | DBH | DPa | Poi | DINH | DB-XII |
---|---|---|---|---|---|---|---|---|

$-L$ | $176.621$ | $186.7$ | $226.355$ | $241.306$ | $202.578$ | $594.751$ | $177.779$ | $198.727$ |

AIC | $355.242$ | $375.4$ | $454.709$ | $486.612$ | $407.155$ | $1191.5$ | $359.558$ | $401.454$ |

CAIC | $355.353$ | $375.511$ | $454.82$ | $486.955$ | $407.267$ | $1191.61$ | $359.901$ | $401.797$ |

BIC | $356.879$ | $377.038$ | $456.347$ | $489.888$ | $408.793$ | $1193.14$ | $362.833$ | $404.729$ |

HQIC | $355.824$ | $375.983$ | $455.292$ | $487.778$ | $407.738$ | $1192.09$ | $360.723$ | $402.619$ |

KS | $0.1620$ | $0.309$ | $0.644$ | $0.779$ | $0.379$ | $0.519$ | $0.171$ | $0.367$ |

p-value | $0.271$ | $0.001$ | <0.001 | <0.001 | <0.001 | <0.001 | 0.245 | <0.001 |

Technique | $\mathit{\beta}$ | KS | p-Value |
---|---|---|---|

ME | $0.945$ | $0.162$ | $0.271$ |

PE | $0.883$ | $0.353$ | $0.0002$ |

Approach | $\mathit{E}\left(\mathit{X}\right)$ | $\mathit{Var}\left(\mathit{X}\right)$ | $\mathit{D}\left(\mathit{X}\right)$ | $\mathit{Sk}\left(\mathit{X}\right)$ | $\mathit{Ku}\left(\mathit{X}\right)$ |
---|---|---|---|---|---|

MLE | $34.923$ | $627.491$ | $17.968$ | $1.414$ | $5.999$ |

PE | $15.561$ | $129.054$ | $8.2937$ | $1.413$ | $5.996$ |

ME | $34.921$ | $627.409$ | $17.967$ | $1.414$ | $5.999$ |

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## Share and Cite

**MDPI and ACS Style**

El-Morshedy, M.; Shahen, H.S.; Almohaimeed, B.; Eliwa, M.S.
Discrete Single-Factor Extension of the Exponential Distribution: Features and Modeling. *Axioms* **2022**, *11*, 737.
https://doi.org/10.3390/axioms11120737

**AMA Style**

El-Morshedy M, Shahen HS, Almohaimeed B, Eliwa MS.
Discrete Single-Factor Extension of the Exponential Distribution: Features and Modeling. *Axioms*. 2022; 11(12):737.
https://doi.org/10.3390/axioms11120737

**Chicago/Turabian Style**

El-Morshedy, Mahmoud, Hend S. Shahen, Bader Almohaimeed, and Mohamed S. Eliwa.
2022. "Discrete Single-Factor Extension of the Exponential Distribution: Features and Modeling" *Axioms* 11, no. 12: 737.
https://doi.org/10.3390/axioms11120737