A Composite Half-Normal-Pareto Distribution with Applications to Income and Expenditure Data
Abstract
:1. Introduction
2. CHNP Distribution
2.1. Density Function
2.2. Properties
- 1.
- The survival function , which is the probability that an article will not fail before time t, is defined by . The survival function for a random variable is given by
- 2.
- The hazards function , defined by , for a random variable, is given by
Right Tail of the CHNP Distribution
Algorithm 1 The algorithm for simulating from the can proceed as follows |
|
- 1:
- Generate
- 2:
- 3:
- Compute
2.3. Actuarial Measure
3. Parameter Estimation
3.1. A Method Based on Percentiles
3.2. ML Estimation
- (i)
- ,
- (ii)
- ,
3.3. Simulation Study
4. Applications
4.1. Numerical Application
- A method based on percentiles. For this example, . We have and . Thus, the estimator of is , where and .
- ML estimation. Using the value of found with the previous method, we calculate the estimator given in Equation (8). This gives us . We observe that the ratio is not satisfied. Therefore, we must use another value for m, for example ; with this, we obtain , and now we observe that the ratio is satisfied, where and . Thus, the ML estimate of is .
4.2. Application to Income Data
4.3. Application to Expenditure Data
5. Concluding Remarks
- The CHNP model has a heavy right tail, as is shown by Proposition 4.
- The support of the CHNP model contains zero. It is a property that is very important for modeling datasets containing zero.
- Cdf, risk function, and quantile function are explicit and are represented by known functions.
- The VaR risk measure is explicit in the CHNP model; in the applications with real data, it is compared with the VaR risk measure of the CEP and Pareto models.
- The applications with income and expenditure data show that the CHNP model provides a better fit than the CEP and Pareto models; it is also observed that the VaR of the CHNP model is closer to the empirical VaR than the VaR of the CEP and Pareto models.
- One reviewer noted the importance of performing a comparison of estimation methods, including Bayesian inference. As we have Fisher’s information for the parameter , we can use it in Jeffrey’s prior to perform Bayesian inference. However, a detailed investigation of the performance of Bayesian estimation is beyond the scope of the present paper.
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Increasing | Decreasing | |
---|---|---|
1 | ||
5 |
Model | p | VaRp | Model | p | VaRp |
---|---|---|---|---|---|
CHNP(0.5) | CHNP(5) | ||||
CHNP(0.5) | CHNP(5) | ||||
CHNP(0.5) | CHNP(5) | ||||
CHNP(1) | CHNP(8) | ||||
CHNP(1) | CHNP(8) | ||||
CHNP(1) | CHNP(8) | 47,281.07 | |||
CHNP(3) | CHNP(10) | ||||
CHNP(3) | CHNP(10) | ||||
CHNP(3) | CHNP(10) | 59,101.34 |
n | Bias | SE | RMSE | CP | |
---|---|---|---|---|---|
1 | 50 | 0.0420 | 0.2492 | 0.2632 | 0.9398 |
100 | 0.0157 | 0.1720 | 0.1758 | 0.9456 | |
150 | 0.0130 | 0.1402 | 0.1433 | 0.9458 | |
200 | 0.0130 | 0.1402 | 0.1227 | 0.9458 | |
2 | 50 | 0.0695 | 0.4952 | 0.5255 | 0.9391 |
100 | 0.0337 | 0.3443 | 0.3564 | 0.9442 | |
150 | 0.0216 | 0.2791 | 0.2863 | 0.9420 | |
200 | 0.0142 | 0.2408 | 0.2449 | 0.9454 | |
3 | 50 | 0.1067 | 0.7441 | 0.7928 | 0.9381 |
100 | 0.0631 | 0.5176 | 0.5361 | 0.9447 | |
150 | 0.0373 | 0.4193 | 0.4340 | 0.9411 | |
200 | 0.0242 | 0.3617 | 0.3700 | 0.9415 | |
4 | 50 | 0.0242 | 0.3617 | 1.0656 | 0.9415 |
100 | 0.0802 | 0.6897 | 0.7206 | 0.9413 | |
150 | 0.0593 | 0.5599 | 0.5703 | 0.9468 | |
200 | 0.0361 | 0.4829 | 0.4930 | 0.9465 |
0.01445457 | 0.01925126 | 0.04748003 | 0.12887746 | 0.19892610 | 0.53610582 |
0.70568085 | 0.72404104 | 0.78969039 | 0.82392295 | 0.84496216 | 0.90191984 |
0.92244296 | 1.21130369 | 1.26907096 | 1.28636763 | 1.30668599 | 1.35093138 |
1.42432950 | 1.65535034 | 1.72375311 | 1.84486889 | 1.96947573 | 2.10059341 |
2.14786935 | 2.68653223 | 2.71026918 | 2.73182813 | 3.10511668 | 3.41038988 |
3.57082832 | 4.44431142 | 5.09754874 | 5.21285944 | 5.60614295 | 6.62777414 |
8.60098244 | 9.32670082 | 10.85377372 | 14.28964739 | 20.51202824 | 25.16157611 |
27.74187053 | 60.16026763 | 65.41449211 | 71.33535927 | 87.67022926 | 102.51836463 |
183.21141945 | 215.76829133 |
n | Median | Mean | Variance | CS | CK |
---|---|---|---|---|---|
50 | 2.417 | 19.474 | 1936.553 | 0.651 | 6.477 |
Model | ML Estimates | AIC | BIC |
---|---|---|---|
CHNP() | 325.641 | 329.855 | |
Pareto() | 378.785 | 380.697 |
n | Median | Mean | Variance | CS | CK |
---|---|---|---|---|---|
500 | 21.125 | 216.709 | 11,270,001 | 0.435 | 1.655 |
Model | ML Estimates | AIC | BIC |
---|---|---|---|
CHNP() | 4937.446 | 4947.875 | |
CEP() | 5049.790 | 5054.005 | |
Pareto() | |||
5867.910 | 5876.339 |
Model∖Significance | ||||||
---|---|---|---|---|---|---|
CHNP() | 25.086 | 40.565 | 75.379 | 180.519 | 803.325 | 3574.872 |
CEP() | 31.813 | 60.186 | 136.919 | 436.094 | 3159.847 | 22,895.580 |
Pareto() | 3.744 | 13.805 | 74.248 | 795.165 | 45,800.120 | 2,638,007 |
n | Median | Mean | Variance | CS | CK |
---|---|---|---|---|---|
75 | 2.5608 | 4.9559 | 87.030 | 0.079 | 1.165 |
Model | ML Estimates | AIC | BIC |
---|---|---|---|
CHNP() | 392.609 | 399.244 | |
CEP() | 405.183 | 411.818 |
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Olmos, N.M.; Gómez-Déniz, E.; Venegas, O.; Gómez, H.W. A Composite Half-Normal-Pareto Distribution with Applications to Income and Expenditure Data. Mathematics 2024, 12, 1631. https://doi.org/10.3390/math12111631
Olmos NM, Gómez-Déniz E, Venegas O, Gómez HW. A Composite Half-Normal-Pareto Distribution with Applications to Income and Expenditure Data. Mathematics. 2024; 12(11):1631. https://doi.org/10.3390/math12111631
Chicago/Turabian StyleOlmos, Neveka M., Emilio Gómez-Déniz, Osvaldo Venegas, and Héctor W. Gómez. 2024. "A Composite Half-Normal-Pareto Distribution with Applications to Income and Expenditure Data" Mathematics 12, no. 11: 1631. https://doi.org/10.3390/math12111631
APA StyleOlmos, N. M., Gómez-Déniz, E., Venegas, O., & Gómez, H. W. (2024). A Composite Half-Normal-Pareto Distribution with Applications to Income and Expenditure Data. Mathematics, 12(11), 1631. https://doi.org/10.3390/math12111631