Extended Exponential Regression Model: Diagnostics and Application to Mineral Data
Abstract
:1. Introduction
 EE$(\alpha ,\beta =0)$ = E$\left(\alpha \right)$.
 EE$(\alpha ,\beta =1)$ = L$\left(\alpha \right)$.
 $\underset{\beta \to +\infty}{lim}$ EE$(\alpha ,\beta )$ = G$(2,\alpha )$.
2. A EE Distribution Parameterized by Its Mean and Mixture Parameters
3. REE Regression Model
3.1. EM Algorithm
 $\mathbb{E}\left(\right)open="("\; close=")">{X}_{i}\mid {\mathit{D}}_{obs};\mathit{\psi}=\frac{\pi {\mu}_{i}/\left({y}_{i}(2\pi )\right)}{1\pi +\pi {\mu}_{i}/\left({y}_{i}(2\pi )\right)}$, $i=1,\dots ,n$.
Algorithm 1 EM algorithm for REE regression model 

3.2. Diagnostic Analysis
Case Deletion Measures
The Hessian Matrix ${\ddot{Q}}_{\mathit{\psi}}\left(\mathit{\psi}\right)$
3.3. Perturbation Schemes
3.3.1. Case Weights Perturbation
3.3.2. Response Perturbation
3.3.3. Covariate Perturbation
3.4. Residual Analysis
4. Simulation Study
5. Applications
5.1. Exploratory Data Analysis to the Mineral Data Set
5.2. Estimation and Model Checking
5.3. Diagnostic Analysis
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
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Skewness  Kurtosis  

RGA  $\frac{2}{\sqrt{\varphi}}$  $3+\frac{6}{\varphi}$ 
RBS  $\frac{4(3\varphi +11)}{{(2\varphi +5)}^{3/2}}$  $\frac{3(41\varphi +186)}{{(2\varphi +5)}^{2}}$ 
REE  $\frac{2(2{\pi}^{3})}{{(2{\pi}^{2})}^{3/2}}$  $\frac{3(8{\pi}^{4}4{\pi}^{2})}{{(2{\pi}^{2})}^{2}}$ 
True Values  $\mathit{n}=50$  $\mathit{n}=100$  

$\mathit{\pi}$  ${\mathit{\gamma}}_{\mathit{0}}$  ${\mathit{\gamma}}_{\mathit{1}}$  ${\mathit{\gamma}}_{\mathit{2}}$  Estimator  Bias  SE  RMSE  CP  Bias  SE  RMSE  CP 
0.2  1  0.5  0.01  $\pi $  −0.018  0.200  0.165  0.992  −0.012  0.125  0.124  0.946 
${\gamma}_{0}$  −0.018  0.158  0.161  0.943  −0.010  0.111  0.112  0.950  
${\gamma}_{1}$  −0.007  0.224  0.226  0.949  0.001  0.156  0.155  0.951  
${\gamma}_{2}$  −0.002  0.116  0.119  0.946  −0.001  0.079  0.080  0.949  
1  −1  −0.01  $\pi $  −0.018  0.197  0.167  0.992  −0.011  0.125  0.122  0.951  
${\gamma}_{0}$  −0.017  0.158  0.164  0.941  −0.009  0.111  0.111  0.951  
${\gamma}_{1}$  0.001  0.224  0.228  0.945  0.001  0.157  0.155  0.952  
${\gamma}_{2}$  −0.002  0.115  0.117  0.949  −0.001  0.080  0.081  0.944  
2  −0.5  0.02  $\pi $  −0.021  0.197  0.164  0.989  −0.010  0.125  0.123  0.949  
${\gamma}_{0}$  −0.019  0.158  0.162  0.940  −0.009  0.110  0.113  0.945  
${\gamma}_{1}$  0.003  0.224  0.228  0.945  0.000  0.157  0.158  0.944  
${\gamma}_{2}$  −0.001  0.115  0.117  0.948  0.000  0.079  0.079  0.948  
0.5  1  0.5  0.01  $\pi $  −0.060  0.265  0.245  0.881  −0.025  0.198  0.190  0.908 
${\gamma}_{0}$  −0.026  0.175  0.186  0.933  −0.010  0.124  0.127  0.943  
${\gamma}_{1}$  0.004  0.250  0.258  0.938  −0.002  0.176  0.178  0.945  
${\gamma}_{2}$  0.001  0.129  0.135  0.936  0.000  0.090  0.091  0.943  
1  −1  −0.01  $\pi $  −0.062  0.267  0.247  0.882  −0.024  0.198  0.189  0.912  
${\gamma}_{0}$  −0.025  0.175  0.184  0.934  −0.013  0.124  0.126  0.945  
${\gamma}_{1}$  0.002  0.249  0.260  0.932  0.003  0.176  0.178  0.946  
${\gamma}_{2}$  0.000  0.129  0.136  0.937  0.002  0.090  0.091  0.947  
2  −0.5  0.02  $\pi $  −0.062  0.267  0.248  0.883  −0.027  0.198  0.189  0.913  
${\gamma}_{0}$  −0.024  0.175  0.186  0.932  −0.011  0.124  0.127  0.943  
${\gamma}_{1}$  0.002  0.249  0.258  0.938  0.001  0.176  0.179  0.945  
${\gamma}_{2}$  0.000  0.129  0.135  0.938  −0.001  0.090  0.093  0.941  
0.75  1  0.5  0.01  $\pi $  −0.136  0.330  0.287  0.857  −0.075  0.267  0.218  0.893 
${\gamma}_{0}$  −0.031  0.187  0.203  0.928  −0.017  0.133  0.140  0.939  
${\gamma}_{1}$  0.002  0.266  0.281  0.934  0.002  0.189  0.199  0.938  
${\gamma}_{2}$  0.001  0.138  0.148  0.935  −0.001  0.096  0.098  0.946  
1  −1  −0.01  $\pi $  −0.138  0.331  0.288  0.853  −0.078  0.266  0.220  0.886  
${\gamma}_{0}$  −0.028  0.186  0.202  0.927  −0.013  0.133  0.138  0.941  
${\gamma}_{1}$  −0.002  0.265  0.283  0.928  −0.002  0.189  0.195  0.939  
${\gamma}_{2}$  0.001  0.138  0.147  0.936  0.000  0.096  0.099  0.944  
2  −0.5  0.02  $\pi $  −0.140  0.328  0.290  0.849  −0.071  0.269  0.216  0.892  
${\gamma}_{0}$  −0.031  0.186  0.203  0.928  −0.015  0.133  0.140  0.935  
${\gamma}_{1}$  0.007  0.265  0.284  0.931  0.006  0.189  0.198  0.938  
${\gamma}_{2}$  0.001  0.138  0.146  0.935  −0.001  0.096  0.100  0.943 
True Values  $\mathit{n}=200$  $\mathit{n}=500$  

$\mathit{\pi}$  ${\mathit{\gamma}}_{\mathit{0}}$  ${\mathit{\gamma}}_{\mathit{1}}$  ${\mathit{\gamma}}_{\mathit{2}}$  Estimator  Bias  SE  RMSE  CP  Bias  SE  RMSE  CP 
0.2  1  0.5  0.01  $\pi $  −0.005  0.087  0.088  0.928  −0.001  0.056  0.056  0.943 
${\gamma}_{0}$  −0.004  0.078  0.079  0.949  −0.001  0.049  0.049  0.949  
${\gamma}_{1}$  0.000  0.110  0.111  0.949  −0.001  0.069  0.070  0.951  
${\gamma}_{2}$  0.001  0.056  0.055  0.953  −0.001  0.035  0.035  0.949  
1  −1  −0.01  $\pi $  −0.005  0.087  0.088  0.929  −0.002  0.056  0.056  0.941  
${\gamma}_{0}$  −0.004  0.078  0.078  0.951  −0.002  0.049  0.049  0.947  
${\gamma}_{1}$  0.000  0.110  0.110  0.950  0.000  0.069  0.069  0.950  
${\gamma}_{2}$  0.001  0.055  0.055  0.946  0.001  0.035  0.035  0.947  
2  −0.5  0.02  $\pi $  −0.007  0.087  0.088  0.928  −0.002  0.056  0.055  0.943  
${\gamma}_{0}$  −0.005  0.078  0.079  0.947  −0.002  0.049  0.049  0.951  
${\gamma}_{1}$  0.000  0.110  0.111  0.950  −0.001  0.069  0.069  0.949  
${\gamma}_{2}$  0.000  0.056  0.056  0.949  0.000  0.035  0.035  0.949  
0.5  1  0.5  0.01  $\pi $  −0.012  0.141  0.140  0.934  −0.004  0.087  0.086  0.953 
${\gamma}_{0}$  −0.006  0.088  0.089  0.946  −0.003  0.056  0.057  0.946  
${\gamma}_{1}$  0.000  0.124  0.125  0.951  0.001  0.079  0.079  0.950  
${\gamma}_{2}$  0.001  0.063  0.064  0.944  0.001  0.039  0.040  0.945  
1  −1  −0.01  $\pi $  −0.009  0.142  0.139  0.933  −0.004  0.087  0.088  0.950  
${\gamma}_{0}$  −0.005  0.088  0.088  0.948  −0.003  0.056  0.056  0.948  
${\gamma}_{1}$  −0.001  0.125  0.125  0.949  0.001  0.079  0.079  0.948  
${\gamma}_{2}$  0.000  0.063  0.062  0.947  0.000  0.040  0.040  0.947  
2  −0.5  0.02  $\pi $  −0.012  0.140  0.137  0.934  −0.004  0.087  0.087  0.950  
${\gamma}_{0}$  −0.006  0.088  0.089  0.946  −0.003  0.056  0.056  0.947  
${\gamma}_{1}$  −0.001  0.124  0.126  0.946  0.001  0.079  0.078  0.950  
${\gamma}_{2}$  −0.001  0.063  0.063  0.948  0.000  0.040  0.040  0.950  
0.75  1  0.5  0.01  $\pi $  −0.041  0.208  0.169  0.913  −0.016  0.141  0.123  0.927 
${\gamma}_{0}$  −0.007  0.095  0.097  0.941  −0.003  0.060  0.062  0.944  
${\gamma}_{1}$  0.001  0.134  0.138  0.941  0.001  0.085  0.086  0.945  
${\gamma}_{2}$  0.000  0.068  0.069  0.945  −0.001  0.043  0.043  0.946  
1  −1  −0.01  $\pi $  −0.039  0.207  0.169  0.914  −0.013  0.142  0.121  0.928  
${\gamma}_{0}$  −0.008  0.095  0.097  0.942  −0.003  0.060  0.061  0.951  
${\gamma}_{1}$  0.002  0.134  0.138  0.940  0.001  0.085  0.085  0.953  
${\gamma}_{2}$  −0.001  0.068  0.069  0.945  0.000  0.043  0.044  0.947  
2  −0.5  0.02  $\pi $  −0.038  0.209  0.168  0.916  −0.014  0.144  0.121  0.931  
${\gamma}_{0}$  −0.007  0.095  0.097  0.943  −0.003  0.060  0.061  0.948  
${\gamma}_{1}$  0.000  0.134  0.137  0.945  −0.001  0.085  0.087  0.946  
${\gamma}_{2}$  0.001  0.068  0.070  0.941  0.000  0.043  0.044  0.945 
${\mathit{y}}_{\left(\mathit{1}\right)}$  MD  $\overline{\mathit{y}}$  SD  CV  CS  CK  ${\mathit{y}}_{\left(\mathit{n}\right)}$  $\mathit{n}$ 
1.00  114.50  133.79  104.46  78.82  0.61  2.58  459.00  86 
Fitted Models  

Parameter  RGA  RBS  REE 
${\widehat{\gamma}}_{0}$  5.0734 (0.1252)  5.0271 (0.1767)  5.0440 (0.1122) 
${\widehat{\gamma}}_{1}$  −0.0145 (0.0060)  −0.0172 (0.0063)  −0.0125 (0.0044) 
pvalue  [0.0148]  [0.0068]  [0.0049] 
$\widehat{\varphi}$  1.1955 (0.1630)     
$\widehat{\xi}$    0.8888 (0.1376)   
$\widehat{\pi}$      0.4650 (0.1713) 
loglikelihood  −503.3155  −520.3540  −502.5457 
AIC  1012.6311  1046.7080  1011.0914 
BIC  1019.9941  1054.0711  1018.4544 
Fitted Models  

Parameter  RGA  RBS  REE 
${\widehat{\gamma}}_{0}$  5.3248 (0.1240)  5.3260 (0.2020)  5.3008 (0.1276) 
${\widehat{\gamma}}_{1}$  −0.0399 (0.0066)  −0.0453 (0.0101)  −0.0381 (0.0069) 
pvalue  [<0.0001]  [<0.0001]  [<0.0001] 
$\widehat{\varphi}$  1.3865 (0.1925)     
$\widehat{\xi}$    1.0595 (0.1640)   
$\widehat{\pi}$      0.3056 (0.1548) 
loglikelihood  −488.7114  −508.1733  −488.1657 
AIC  983.4229  1022.3466  982.3314 
BIC  990.7508  1029.6746  989.6594 
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Gómez, Y.M.; Gallardo, D.I.; Leão, J.; Gómez, H.W. Extended Exponential Regression Model: Diagnostics and Application to Mineral Data. Symmetry 2020, 12, 2042. https://doi.org/10.3390/sym12122042
Gómez YM, Gallardo DI, Leão J, Gómez HW. Extended Exponential Regression Model: Diagnostics and Application to Mineral Data. Symmetry. 2020; 12(12):2042. https://doi.org/10.3390/sym12122042
Chicago/Turabian StyleGómez, Yolanda M., Diego I. Gallardo, Jeremias Leão, and Héctor W. Gómez. 2020. "Extended Exponential Regression Model: Diagnostics and Application to Mineral Data" Symmetry 12, no. 12: 2042. https://doi.org/10.3390/sym12122042