A New Two-Parameter Discrete Distribution for Overdispersed and Asymmetric Data: Its Properties, Estimation, Regression Model, and Applications
Abstract
:1. Introduction
- The first thing that needs to be accomplished is to create a new two-parameter Poisson transmuted moment exponential distribution. This may be performed by combining the Poisson distribution with the transmuted moment exponential distribution. When compared to existing discrete distributions, the moments and related measures of the new model may be determined analytically, and it possesses a high modeling capability. Additionally, the new model is extremely flexible.
- The model parameters are estimated using the renowned maximum likelihood estimation approach and a comprehensive simulation study to illustrate the pattern of these derived ML estimators.
- A new count regression model is also proposed to replace some existing count regression models.
- Two asymmetric datasets from different real-life areas are utilized to show the flexibility of the new distribution over some well-known probability distributions and regression models.
- We also estimate the model parameters using the Bayesian approach.
2. Derivation of New Model
Moments and Associated Measures
3. Parameter Estimation
3.1. Maximum Likelihood Estimation
3.2. Bayesian Estimation
Metropolis–Hastings (M-H) algorithm
- Start with the initial parameter values .
- Set the iteration counter to .
- Simulate the and from the normal proposal distribution and , respectively.
- Then, evaluate the acceptance probability:
- 5
- Then, generate and from the uniform distribution .
- 6
- If , we consider ; otherwise, set .
- 7
- If , we consider ; otherwise, set .
- 8
- Change the counter from to .
- 9
- To obtain an accurate approximation for the estimates, we must repeat the procedures from (3)–(8). repetitions to obtain values for the parameters, and this sample can be stated as follows: .
4. Simulation
5. PTMEx Regression Model
6. Empirical Study
6.1. European Corn Borer Data
6.2. Length of Hospital Stay Data
- length of patients’ stays at the hospital.
- cardiovascular procedure (1 = CABG, 0 = PTCA).
- gender (1 = male, 0 = female).
- admission type (1 = urgent, 0 = elective).
- age (1 = age > 75, 0 = age ≤ 75).
7. Conclusions
8. Future Work
- Estimation methods: Future work could focus on developing efficient and accurate estimation methods for the parameters of the proposed distribution. This could involve maximum likelihood estimation, Bayesian estimation, or robust estimation techniques. Researchers may also explore the properties of the estimators, such as their asymptotic behavior and efficiency.
- Regression modeling: The new discrete distribution could be incorporated into regression models to analyze its performance in predicting or explaining the relationships between variables. This could involve developing regression frameworks, such as generalized linear models or zero-inflated models, that utilize the proposed distribution as the response variable. Researchers could also explore model selection criteria and compare the performance of the new distribution with existing ones in regression settings.
- Simulation studies: Future research could involve conducting extensive simulation studies to evaluate the behavior of the proposed distribution under various scenarios. This could include examining its robustness to violations of assumptions, assessing the accuracy of parameter estimation methods, and comparing the performances of statistical tests based on the new distribution.
- Applications: Further exploration of practical applications could be an area of focus. Researchers may investigate real-world datasets with overdispersed and asymmetric characteristics to assess the adequacy of the proposed distribution in modeling such data. This could include applications in fields such as finance, epidemiology, ecology, or social sciences.
- Software development: To facilitate the adoption and usage of the new distribution, researchers may develop software packages or functions in statistical software platforms (e.g., R and Python) for estimating parameters, conducting inference, and implementing regression models based on the proposed distribution. This would make it easier for practitioners to apply the distribution in their own research or data analysis.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Parameters | Measures | ||||||
---|---|---|---|---|---|---|---|
Mean | Variance | Skewness | Kurtosis | CV | DI | ||
0.5 | −0.8 | 1.3000 | 1.8600 | 1.3695 | 5.5696 | 1.0491 | 1.4308 |
−0.5 | 1.1875 | 1.7461 | 1.4609 | 5.9067 | 1.1128 | 1.4704 | |
−0.2 | 1.0750 | 1.6069 | 1.5621 | 6.3336 | 1.1792 | 1.4948 | |
0.0 | 1.0000 | 1.5000 | 1.6330 | 6.6667 | 1.2247 | 1.5000 | |
0.2 | 0.9250 | 1.3819 | 1.7037 | 7.0283 | 1.2708 | 1.4939 | |
0.5 | 0.8125 | 1.1836 | 1.7963 | 7.5506 | 1.3390 | 1.4567 | |
0.8 | 0.7000 | 0.9600 | 1.8244 | 7.6852 | 1.3997 | 1.3714 | |
1.0 | −0.8 | 2.6000 | 4.8400 | 1.2464 | 5.2747 | 0.8462 | 1.8615 |
−0.5 | 2.3750 | 4.6094 | 1.3275 | 5.5368 | 0.9040 | 1.9408 | |
−0.2 | 2.1500 | 4.2775 | 1.4267 | 5.9217 | 0.9620 | 1.9895 | |
0.0 | 2.0000 | 4.0000 | 1.5000 | 6.2500 | 1.0000 | 2.0000 | |
0.2 | 1.8500 | 3.6775 | 1.5751 | 6.6303 | 1.0366 | 1.9878 | |
0.5 | 1.6250 | 3.1094 | 1.6735 | 7.2268 | 1.0851 | 1.9135 | |
0.8 | 1.4000 | 2.4400 | 1.6813 | 7.3870 | 1.1157 | 1.7429 | |
1.5 | −0.8 | 3.9000 | 8.9400 | 1.2105 | 5.2069 | 0.7667 | 2.2923 |
−0.5 | 3.5625 | 8.5898 | 1.2847 | 5.4302 | 0.8227 | 2.4112 | |
−0.2 | 3.2250 | 8.0119 | 1.3840 | 5.7992 | 0.8777 | 2.4843 | |
0.0 | 3.0000 | 7.5000 | 1.4606 | 6.1333 | 0.9129 | 2.5000 | |
0.2 | 2.7750 | 6.8869 | 1.5412 | 6.5368 | 0.9457 | 2.4818 | |
0.5 | 2.4375 | 5.7773 | 1.6499 | 7.2096 | 0.9861 | 2.3702 | |
0.8 | 2.1000 | 4.4400 | 1.6559 | 7.4460 | 1.0034 | 2.1143 |
Parameter | AE | MRE | MSE | ||||
---|---|---|---|---|---|---|---|
50 | 0.5542 | −0.5740 | 0.1083 | −0.2825 | 0.0233 | 0.3661 | |
100 | 0.5448 | −0.6135 | 0.0896 | −0.2332 | 0.0175 | 0.2813 | |
200 | 0.5252 | −0.6892 | 0.0504 | −0.1385 | 0.0084 | 0.1649 | |
500 | 0.5129 | −0.7327 | 0.0259 | −0.0841 | 0.0038 | 0.0923 | |
1000 | 0.5025 | −0.7884 | 0.0049 | −0.0145 | 0.0012 | 0.0395 | |
50 | 0.5262 | −0.4536 | 0.0524 | −0.0928 | 0.0209 | 0.3593 | |
100 | 0.5415 | −0.3757 | 0.0830 | −0.2486 | 0.0198 | 0.3749 | |
200 | 0.5284 | −0.4502 | 0.0567 | −0.0996 | 0.0135 | 0.2745 | |
500 | 0.5222 | −0.4367 | 0.0444 | −0.1266 | 0.0096 | 0.2053 | |
1000 | 0.5058 | −0.4864 | 0.0117 | −0.0273 | 0.0031 | 0.0846 | |
50 | 0.4960 | −0.3877 | 0.0079 | −0.9385 | 0.0199 | 0.4136 | |
100 | 0.5086 | −0.2877 | 0.0172 | −0.4385 | 0.0156 | 0.3724 | |
200 | 0.5186 | −0.2150 | 0.0371 | −0.0751 | 0.0139 | 0.3089 | |
500 | 0.5213 | −0.1594 | 0.0425 | −0.2032 | 0.0096 | 0.2134 | |
1000 | 0.5159 | −0.1697 | 0.0317 | −0.1514 | 0.0066 | 0.1415 | |
50 | 0.5262 | −0.4536 | 0.0524 | −0.0928 | 0.0209 | 0.3593 | |
100 | 0.5415 | −0.3757 | 0.083 | −0.2486 | 0.0198 | 0.3749 | |
200 | 0.5284 | −0.4502 | 0.0567 | −0.0996 | 0.0135 | 0.2745 | |
500 | 0.5222 | −0.4367 | 0.0444 | −0.1266 | 0.0096 | 0.2053 | |
1000 | 0.5058 | −0.4864 | 0.0117 | −0.0273 | 0.0031 | 0.0846 | |
50 | 1.5044 | 0.4211 | 0.0029 | 0.1578 | 0.0998 | 0.2871 | |
100 | 1.4681 | 0.3977 | 0.0213 | 0.2046 | 0.0831 | 0.2145 | |
200 | 1.4856 | 0.4152 | 0.0096 | 0.1695 | 0.0675 | 0.1776 | |
500 | 1.4949 | 0.4434 | 0.0034 | 0.1133 | 0.0509 | 0.1112 | |
1000 | 1.5066 | 0.4720 | 0.0044 | 0.0560 | 0.0383 | 0.0841 | |
50 | 1.5703 | −0.4694 | 0.0469 | −0.0611 | 0.1397 | 0.2699 | |
100 | 1.5543 | −0.4627 | 0.0362 | −0.0747 | 0.0826 | 0.1893 | |
200 | 1.5438 | −0.4605 | 0.0292 | −0.0790 | 0.0633 | 0.1381 | |
500 | 1.5131 | −0.4930 | 0.0088 | −0.0140 | 0.0265 | 0.0659 | |
1000 | 1.5019 | −0.5030 | 0.0013 | −0.0060 | 0.0064 | 0.0209 |
Count | Observed | Expected | |||||
---|---|---|---|---|---|---|---|
PTMEx | DBurr | PMEx | DB | DITL | Poisson | ||
0 | 43 | 40.417 | 33.438 | 39.558 | 32.741 | 52.189 | 27.226 |
1 | 35 | 33.658 | 31.574 | 33.691 | 39.589 | 30.424 | 40.385 |
2 | 17 | 21.052 | 22.360 | 21.521 | 24.275 | 14.112 | 29.951 |
3 | 11 | 11.818 | 14.076 | 12.220 | 12.505 | 7.4663 | 14.809 |
4 | 5 | 6.3101 | 8.3071 | 6.5047 | 5.9678 | 4.3900 | 5.4915 |
5 | 4 | 3.2874 | 4.7064 | 3.3240 | 2.7359 | 2.7906 | 1.6291 |
6 | 1 | 1.6923 | 2.5924 | 1.6514 | 1.2256 | 1.8811 | 0.4027 |
7 | 2 | 0.8660 | 1.3988 | 0.8037 | 0.5414 | 1.3271 | 0.0853 |
8 | 2 | 0.8978 | 1.5473 | 0.7259 | 0.4193 | 5.4195 | 0.0188 |
Total | 120 | 120 | 120 | 120 | 120 | 120 | |
MLE | 0.89444 | 0.51916 | 0.74161 | 2.3767 | 1.9840 | 1.4833 | |
0.46514 | 2.35785 | - | - | - | - | ||
GOF Measures | 200.82 | 204.29 | 201.22 | 204.68 | 205.15 | 219.19 | |
AIC | 405.64 | 412.59 | 404.44 | 411.35 | 412.30 | 440.38 | |
BIC | 411.22 | 418.16 | 407.23 | 414.14 | 415.09 | 443.16 | |
2.0825 | 6.5310 | 2.7268 | 9.6431 | 6.9771 | 21.761 | ||
df | 3.0 | 3.0 | 4.0 | 4.0 | 4.0 | 3.0 | |
p-value | 0.72058 | 0.08845 | 0.60450 | 0.04689 | 0.13710 | <0.0001 |
Para. | P | NB | PQL | PTMEx | ||||
---|---|---|---|---|---|---|---|---|
MLEs (SE) | p-Value | MLEs (SE) | p-Value | MLEs (SE) | p-Value | MLEs (SE) | p-Value | |
1.4560 (0.0158) | <0.0001 | 1.0780 (0.0298) | <0.0001 | 1.3624 (0.0402) | <0.0001 | 1.3907 (0.0331) | <0.0001 | |
0.9603 (0.0122) | <0.0001 | 1.0866 (0.0243) | <0.0001 | 0.9746 (0.0317) | <0.0001 | 0.9877 (0.0260) | <0.0001 | |
−0.1239 (0.0118) | <0.0001 | 0.0724 (0.0249) | 0.0030 | −0.1273 (0.0332) | 0.0001 | −0.1275 (0.0272) | <0.0001 | |
0.3266 (0.0121) | <0.0001 | 0.5319 (0.0249) | <0.0001 | 0.3961 (0.0329) | <0.0001 | 0.3909 (0.0270) | <0.0001 | |
0.1222 (0.0124) | <0.0001 | 0.3161 (0.3161) | <0.0001 | 0.1180 (0.0353) | <0.0001 | 0.1224 (0.0289) | <0.0001 | |
- | - | - | - | 0.8893 (0.0016) | - | 0.9843 (0.0109) | - | |
11,190 | 10,578 | 10,919 | 10,352 | |||||
AIC | 22,390 | 21,169 | 21,849 | 20,714 | ||||
BIC | 22,421 | 21,206 | 21,880 | 20,745 |
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Alrumayh, A.; Khogeer, H.A. A New Two-Parameter Discrete Distribution for Overdispersed and Asymmetric Data: Its Properties, Estimation, Regression Model, and Applications. Symmetry 2023, 15, 1289. https://doi.org/10.3390/sym15061289
Alrumayh A, Khogeer HA. A New Two-Parameter Discrete Distribution for Overdispersed and Asymmetric Data: Its Properties, Estimation, Regression Model, and Applications. Symmetry. 2023; 15(6):1289. https://doi.org/10.3390/sym15061289
Chicago/Turabian StyleAlrumayh, Amani, and Hazar A. Khogeer. 2023. "A New Two-Parameter Discrete Distribution for Overdispersed and Asymmetric Data: Its Properties, Estimation, Regression Model, and Applications" Symmetry 15, no. 6: 1289. https://doi.org/10.3390/sym15061289
APA StyleAlrumayh, A., & Khogeer, H. A. (2023). A New Two-Parameter Discrete Distribution for Overdispersed and Asymmetric Data: Its Properties, Estimation, Regression Model, and Applications. Symmetry, 15(6), 1289. https://doi.org/10.3390/sym15061289