A Multivariate Flexible Skew-Symmetric-Normal Distribution: Scale-Shape Mixtures and Parameter Estimation via Selection Representation
Abstract
:1. Introduction
2. Methodology
2.1. The Family of SSMFSSN Distributions
2.2. Parameter Estimation via the ECME Algorithm
- CMQ-Step 1: Fixing and , we update via Proposition A2 by taking the partial derivative of (22) with respect to . Since the derivation cannot get a closed-form expression for its maximizer, the solution of is validated by numerically solving the root of the following equation:
- CMQ-Step 2: Fixing and then updating by maximizing (22) over gives
- CMQ-Step 3: Fixing , we update via Proposition A3 by taking the partial derivative of (22) with respect to each, . Since their solutions cannot be isolated and set equal to zeros, we have the following equation for finding the nonlinear roots of :
- CML-Step: is updated by optimizing the following constrained log-likelihood function:
3. Examples of SSMFSSN Distributions
3.1. The Multivariate Flexible Skew-Symmetric-t-Normal Distribution
- CMQ-Step 4: is obtained by solving the root of the following equation:
3.2. The Multivariate Flexible Skew-Symmetric-Slash-Normal Distribution
- CMQ-Step 4:
3.3. The Multivariate Flexible Skew-Symmetric-Contaminated-Normal Distribution
3.4. The Multivariate Flexible Skew-Symmetric-t Distribution
3.5. The Multivariate Flexible Skew-Symmetric-t-t Distribution
- CMQ-Step 4: and are obtained by solving the roots of the following two equations:
4. Simulation Studies
4.1. Recovery of the True Underlying Parameters
4.2. Comparing the Proposed Procedure with Convolution-Type EM Algorithms
- CMQ-Step 1: Fixing and , we obtain by
5. An Illustrative Example: The Wind Speed Data
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. The Hadamard Product
Appendix B. Proof of Equation (43)
References
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Model | Parameter | n = 100 | n = 250 | n = 500 | n = 1000 | ||||
---|---|---|---|---|---|---|---|---|---|
MAB | RMSE | MAB | RMSE | MAB | RMSE | MAB | RMSE | ||
MFSSN | 0.090 | 0.122 | 0.057 | 0.074 | 0.037 | 0.048 | 0.027 | 0.033 | |
0.257 | 0.382 | 0.143 | 0.209 | 0.115 | 0.168 | 0.081 | 0.123 | ||
0.476 | 0.625 | 0.203 | 0.270 | 0.111 | 0.155 | 0.079 | 0.112 | ||
0.339 | 0.489 | 0.160 | 0.209 | 0.104 | 0.131 | 0.064 | 0.082 | ||
MFSSTN | 0.095 | 0.127 | 0.062 | 0.086 | 0.040 | 0.053 | 0.028 | 0.036 | |
0.419 | 0.673 | 0.244 | 0.383 | 0.158 | 0.241 | 0.121 | 0.179 | ||
0.382 | 0.469 | 0.349 | 0.426 | 0.317 | 0.382 | 0.306 | 0.355 | ||
0.335 | 0.439 | 0.194 | 0.244 | 0.177 | 0.216 | 0.151 | 0.181 | ||
2.495 | 10.745 | 0.458 | 0.651 | 0.319 | 0.413 | 0.203 | 0.258 | ||
MFSSSLN | 0.091 | 0.121 | 0.052 | 0.069 | 0.036 | 0.047 | 0.025 | 0.035 | |
0.424 | 0.641 | 0.306 | 0.489 | 0.190 | 0.314 | 0.127 | 0.200 | ||
0.458 | 0.588 | 0.267 | 0.347 | 0.216 | 0.269 | 0.174 | 0.206 | ||
0.470 | 0.614 | 0.259 | 0.378 | 0.174 | 0.227 | 0.114 | 0.138 | ||
6.530 | 11.849 | 3.425 | 8.206 | 1.191 | 3.251 | 0.481 | 0.721 | ||
MFSSCNe | 0.087 | 0.117 | 0.048 | 0.064 | 0.036 | 0.047 | 0.026 | 0.034 | |
0.363 | 0.546 | 0.285 | 0.440 | 0.223 | 0.339 | 0.133 | 0.202 | ||
0.446 | 0.595 | 0.288 | 0.353 | 0.202 | 0.247 | 0.138 | 0.174 | ||
0.426 | 0.586 | 0.265 | 0.347 | 0.178 | 0.219 | 0.131 | 0.160 | ||
0.199 | 0.216 | 0.162 | 0.176 | 0.116 | 0.137 | 0.083 | 0.098 | ||
MFSST | 0.114 | 0.152 | 0.069 | 0.091 | 0.042 | 0.055 | 0.032 | 0.040 | |
0.393 | 0.586 | 0.250 | 0.386 | 0.160 | 0.238 | 0.132 | 0.216 | ||
0.453 | 0.559 | 0.265 | 0.329 | 0.207 | 0.253 | 0.194 | 0.228 | ||
0.355 | 0.468 | 0.215 | 0.273 | 0.137 | 0.175 | 0.110 | 0.139 | ||
1.152 | 2.210 | 0.430 | 0.621 | 0.266 | 0.366 | 0.207 | 0.265 | ||
MFSSTTe | 0.114 | 0.146 | 0.070 | 0.094 | 0.047 | 0.061 | 0.029 | 0.038 | |
0.400 | 0.654 | 0.213 | 0.323 | 0.165 | 0.255 | 0.115 | 0.188 | ||
0.437 | 0.546 | 0.411 | 0.476 | 0.406 | 0.453 | 0.414 | 0.438 | ||
0.313 | 0.409 | 0.216 | 0.260 | 0.183 | 0.217 | 0.167 | 0.193 | ||
1.332 | 2.733 | 0.424 | 0.573 | 0.300 | 0.408 | 0.210 | 0.279 |
Family | Model | d | AIC | BIC | |
---|---|---|---|---|---|
MSTC | –3178.7 | 13 | 6383.4 | 6430.5 | |
MSSMSN | MST | –3180.7 | 13 | 6387.5 | 6434.6 |
MSTN | –3180.9 | 13 | 6387.8 | 6434.9 | |
MFSSN | –3171.7 | 15 | 6373.4 | 6427.8 | |
MFSSTN | –3145.6 | 16 | 6323.1 | 6381.2 | |
SSMFSSN | MFSSSLN | –3147.1 | 16 | 6326.2 | 6384.2 |
MFSSCN | –3145.6 | 16 | 6323.3 | 6381.3 | |
MFSST | –3143.0 | 16 | 6318.1 | 6376.1 | |
MFSSTT | –3138.2 | 17 | 6310.4 | 6372.1 |
Parameter | MFSSN | MFSSTN | MFSSSLN | MFSSCN | MFSST | MFSSTT |
---|---|---|---|---|---|---|
23.0(0.05) | 21.3 (0.09) | 21.1 (0.04) | 21.5 (0.07) | 19.9 (0.06) | 18.6 (0.08) | |
14.8 (0.04) | 15.6 (0.08) | 15.2 (0.04) | 15.0 (0.07) | 15.1 (0.04) | 15.0 (0.06) | |
14.6 (0.04) | 14.9 (0.06) | 14.8 (0.02) | 15.4 (0.04) | 13.4 (0.08) | 12.6 (0.09) | |
221.2 (0.26) | 122.9 (0.49) | 80.6 (0.24) | 115.8 (0.31) | 115.6 (0.21) | 115.6 (0.25) | |
138.8 (0.16) | 96.4 (0.36) | 64.6 (0.18) | 91.5 (0.24) | 92.8 (0.15) | 93.2 (0.17) | |
151.0 (0.13) | 104.8 (0.26) | 69.5 (0.13) | 99.1 (0.16) | 102.9 (0.18) | 106.0 (0.19) | |
181.3 (0.10) | 134.5 (0.27) | 90.3 (0.16) | 128.0 (0.18) | 131.2 (0.18) | 132.1 (0.20) | |
111.4 (0.07) | 80.8 (0.18) | 54.4 (0.09) | 79.6 (0.12) | 78.8 (0.15) | 79.1 (0.15) | |
296.4 (0.07) | 203.4 (0.19) | 134.0 (0.13) | 187.2 (0.09) | 204.4 (0.24) | 206.7 (0.27) | |
–1.7 (0.01) | –0.4 (0.04) | –0.2 (0.01) | –0.3 (0.02) | 0.1 (0.02) | 1.0 (0.02) | |
1.6 (0.04) | 1.1 (0.04) | 1.0 (0.03) | 1.3 (0.04) | 1.3 (0.01) | 3.2 (0.01) | |
1.9 (0.04) | 1.4 (0.04) | 1.1 (0.02) | 1.3 (0.02) | 1.4 (0.01) | 2.8 (0.02) | |
–0.7 (1.99) | –0.1 (2.36) | 0.0 (1.03) | –0.1 (1.53) | –0.2 (0.04) | 1.2 (0.32) | |
–0.7 (0.05) | –0.4 (0.25) | –0.2 (0.05) | –0.4 (0.19) | –0.5 (0.01) | –1.6 (0.21) | |
–0.1 (1.52) | –0.2 (1.98) | –0.1 (0.78) | –0.2 (1.18) | –0.1 (0.02) | –2.3 (0.17) | |
– | 5.0 (0.18) | 1.5 (0.03) | 0.2 (0.04) | 4.7 (0.06) | 4.7 (0.08) | |
– | – | – | 0.2 (0.54) | – | 1.0 (0.16) |
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Mahdavi, A.; Amirzadeh, V.; Jamalizadeh, A.; Lin, T.-I. A Multivariate Flexible Skew-Symmetric-Normal Distribution: Scale-Shape Mixtures and Parameter Estimation via Selection Representation. Symmetry 2021, 13, 1343. https://doi.org/10.3390/sym13081343
Mahdavi A, Amirzadeh V, Jamalizadeh A, Lin T-I. A Multivariate Flexible Skew-Symmetric-Normal Distribution: Scale-Shape Mixtures and Parameter Estimation via Selection Representation. Symmetry. 2021; 13(8):1343. https://doi.org/10.3390/sym13081343
Chicago/Turabian StyleMahdavi, Abbas, Vahid Amirzadeh, Ahad Jamalizadeh, and Tsung-I Lin. 2021. "A Multivariate Flexible Skew-Symmetric-Normal Distribution: Scale-Shape Mixtures and Parameter Estimation via Selection Representation" Symmetry 13, no. 8: 1343. https://doi.org/10.3390/sym13081343