The HeavyTailed Gleser Model: Properties, Estimation, and Applications
Abstract
:1. Introduction
2. The G Distribution
2.1. Properties
 (a)
 the G distribution has unimodality (at 0).
 (b)
 $\frac{\sigma Y}{1Y}\sim G(\sigma ,\alpha )$.
 (c)
 the cumulative distribution function (cdf) of X is given by$${F}_{X}(x;\sigma ,\alpha )={I}_{y}\left(\right)open="("\; close=")">\overline{\alpha},\alpha $$
 (d)
 the hazard function of X is decreasing for all $x>0$.
 (e)
 the rth moment of the random variable X does not exist for $r\ge \alpha $.
 (f)
 $\frac{1}{1+X}\sim Beta(\alpha ,\overline{\alpha}).$
 (g)
 the quantile function (Q) of the G distribution is given by$$\begin{array}{c}\hfill Q\left(p\right)=\frac{\sigma {I}_{p}^{1}(\overline{\alpha},\alpha )}{1{I}_{p}^{1}(\overline{\alpha},\alpha )},\phantom{\rule{2.em}{0ex}}0<p<1,\end{array}$$
 (e)
 Considering $\sigma =1$, we claim that the integral ${\int}_{1}^{\infty}\frac{{x}^{\alpha +r}}{1+x}dx$ is divergent.In fact, taking ${g}_{1}\left(x\right)=\frac{{x}^{\alpha +r}}{1+x}$ and ${g}_{2}\left(x\right)=\frac{1}{{x}^{1+\alpha r}}$, we have that$$\underset{x\to \infty}{lim}\frac{{g}_{1}\left(x\right)}{{g}_{2}\left(x\right)}=\underset{x\to \infty}{lim}\frac{x}{1+x}=1\ne 0;$$On the other hand, since $\mathbb{E}\left({X}^{r}\right)=\frac{1}{B(\overline{\alpha},\alpha )}{\int}_{0}^{\infty}\frac{{x}^{\alpha +r}}{1+x}dx$, by using comparison$$0\le {\int}_{1}^{\infty}\frac{{x}^{\alpha +r}}{1+x}dx\le {\int}_{0}^{\infty}\frac{{x}^{\alpha +r}}{1+x}dx,$$
2.2. Actuarial Measure
2.3. Order Statistics
2.4. Entropy
2.4.1. Shannon Entropy
 1.
 $\mathbb{E}\left(\mathrm{log}X\right)=log\sigma +\psi \left(\overline{\alpha}\right)\psi \left(\alpha \right),$
 2.
 $\mathbb{E}\left(\mathrm{log}\right(\sigma +X\left)\right)=log\sigma \gamma \psi \left(\alpha \right),$
2.4.2. Rényi Entropy
3. Tail of the Distribution
4. Inference
4.1. ML Estimation
4.2. Simulation Study
Algorithm 1 For simulating from the $X\sim G(\sigma ,\alpha )$ can proceed as follows 

Algorithm 2 For simulating from the $X\sim G(\sigma ,\alpha )$ can proceed as follows 

4.3. Fisher’s Information Matrix
5. Applications
 $f(x;\alpha ,\beta )=\frac{\beta {\alpha}^{\beta}}{{x}^{\beta +1}},\phantom{\rule{1.em}{0ex}}x>\alpha ,$
 $f(x;\sigma ,\alpha )=\frac{2\alpha {x}^{\alpha 1}}{{\sigma}^{\alpha}}\varphi \left(\right)open="("\; close=")">{\left(\right)}^{\frac{x}{\sigma}}\alpha ,\phantom{\rule{1.em}{0ex}}x0,$
5.1. Numerical Application
5.2. Application with Real Data
6. Discussion
 The smaller parameter $\alpha $, the heavier the right tail of the G model.
 The G model has an explicit representation given in Proposition 1 $\left(b\right)$.
 Cdf, risk function and quantile function are explicit and are represented by known functions.
 The VaR measurement is explicit and is used to show that the right tail of the G model is heavy.
 The applications show that the G model has its own characteristic compared with other twoparameter models, and that the G model can be a good candidate for modelling income data with a heavy tail.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
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Distribution  $\mathit{P}\phantom{\rule{0.166667em}{0ex}}(\mathit{X}>2)$  $\mathit{P}\phantom{\rule{0.166667em}{0ex}}(\mathit{X}>3)$  $\mathit{P}\phantom{\rule{0.166667em}{0ex}}(\mathit{X}>4)$ 

G(1, 0.9)  $0.04803$  $0.03537$  $0.02818$ 
G(1, 0.7)  $0.19065$  $0.15106$  $0.12697$ 
G(1, 0.3)  $0.63376$  $0.57717$  $0.53760$ 
True Value  $\mathit{n}=100$  $\mathit{n}=200$  $\mathit{n}=500$  

$\mathit{\sigma}$  $\mathit{\alpha}$  Par.  Bias  SE  RMSE  CP  Bias  SE  RMSE  CP  Bias  SE  RMSE  CP 
1  0.2  $\sigma $  0.1934  0.5709  0.6543  0.9402  0.1113  0.3727  0.4083  0.9474  0.0494  0.2219  0.2292  0.9540 
$\alpha $  0.0054  0.0260  0.0270  0.9618  0.0039  0.0181  0.0186  0.9562  0.0022  0.0113  0.0116  0.9542  
0.5  $\sigma $  0.2191  0.7544  0.8668  0.9004  0.1006  0.4920  0.5425  0.9156  0.0468  0.3010  0.3131  0.9370  
$\alpha $  0.0004  0.0679  0.0667  0.9258  $0.0012$  0.0496  0.0491  0.9308  0.0005  0.0322  0.0320  0.9424  
0.8  $\sigma $  0.0727  0.5090  0.5498  0.8988  0.0348  0.3463  0.3602  0.9216  0.0118  0.2137  0.2143  0.9432  
$\alpha $  $0.0044$  0.0258  0.0271  0.9550  $0.0022$  0.0179  0.0183  0.9496  $0.0010$  0.0112  0.0114  0.9510  
10  0.2  $\sigma $  1.9401  5.7013  6.4320  0.9394  1.1103  3.7292  4.0676  0.9486  0.4651  2.2140  2.2556  0.9528 
$\alpha $  0.0052  0.0260  0.0268  0.9562  0.0037  0.0181  0.0189  0.9492  0.0020  0.0113  0.0112  0.9560  
0.5  $\sigma $  1.9611  7.4418  8.4714  0.8872  1.0610  4.9654  5.2373  0.9176  0.4148  2.9938  3.0764  0.9360  
$\alpha $  $0.0014$  0.0681  0.0668  0.9216  0.0002  0.0497  0.0490  0.9344  0.0000  0.0322  0.0315  0.9484  
0.8  $\sigma $  0.6832  5.0843  5.3895  0.8952  0.3724  3.4679  3.5625  0.9236  0.1692  2.1465  2.1721  0.9406  
$\alpha $  $0.0047$  0.0259  0.0274  0.9510  $0.0020$  0.0179  0.0181  0.9544  $0.0008$  0.0112  0.0113  0.9534 
Model  ML Estimates  AIC  BIC 

G($\sigma ,\alpha $)  $\widehat{\sigma}=0.890$, $\widehat{\alpha}=0.295$  1967.556  1974.152 
Pareto($\alpha ,\beta $)  $\widehat{\alpha}=0.002$, $\widehat{\beta}=0.121$  2144.958  2151.555 
GHN($\sigma ,\alpha $)  $\widehat{\sigma}=204.871$, $\widehat{\alpha}=0.144$  2143.358  2151.787 
n  Median  Mean  Variance  CS  CK 

500  21.125  216.709  11,270,001  0.435  1.655 
Model  ML Estimates  AIC  BIC 

G($\sigma ,\alpha $)  $\widehat{\sigma}=21.555\phantom{\rule{0.166667em}{0ex}}\left(6.264\right)$, $\widehat{\alpha}=0.497\phantom{\rule{0.166667em}{0ex}}\left(0.033\right)$  5139.176  5147.605 
Pareto($\alpha ,\beta $)  $\widehat{\alpha}=0.065\phantom{\rule{0.166667em}{0ex}}\left(0.001\right)$, $\widehat{\beta}=0.171\phantom{\rule{0.166667em}{0ex}}\left(0.008\right)$  5867.910  5876.339 
GHN($\sigma ,\alpha $)  $\widehat{\sigma}=64.313\phantom{\rule{0.166667em}{0ex}}\left(6.815\right)$, $\widehat{\alpha}=0.335\phantom{\rule{0.166667em}{0ex}}\left(0.008\right)$  5280.360  5288.789 
Model∖Significance  0.50 (21.125)  0.60 (28.600)  0.70 (40.000)  0.80 (60.000)  0.90 (100.000)  0.95 (210.250) 

G ($\widehat{\sigma},\widehat{\alpha}$)  22.034  41.764  85.060  209.891  889.770  3632.538 
Pareto ($\widehat{\alpha},\widehat{\beta}$)  3.744  13.805  74.248  795.165  45800.120  2638007 
GHN ($\widehat{\sigma},\widehat{\alpha}$)  19.836  38.426  71.568  134.928  284.355  479.988 
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Olmos, N.M.; GómezDéniz, E.; Venegas, O. The HeavyTailed Gleser Model: Properties, Estimation, and Applications. Mathematics 2022, 10, 4577. https://doi.org/10.3390/math10234577
Olmos NM, GómezDéniz E, Venegas O. The HeavyTailed Gleser Model: Properties, Estimation, and Applications. Mathematics. 2022; 10(23):4577. https://doi.org/10.3390/math10234577
Chicago/Turabian StyleOlmos, Neveka M., Emilio GómezDéniz, and Osvaldo Venegas. 2022. "The HeavyTailed Gleser Model: Properties, Estimation, and Applications" Mathematics 10, no. 23: 4577. https://doi.org/10.3390/math10234577