An Extension of the Truncated-Exponential Skew- Normal Distribution
Abstract
:1. Introduction
2. Incorporating Kurtosis
2.1. Representation
2.2. Probability Density Function
2.3. Reliability Analysis
2.4. Moments
2.5. Incorporation of Parameters
2.6. Log Likelihood Equations
2.7. STESN or TESN Model?
3. Simulation Study
4. Application to a Data Set
5. Final Comments
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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True Values | ||||||||||||||||||||
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q | bias | SE | RMSE | CP | bias | SE | RMSE | CP | bias | SE | RMSE | CP | bias | SE | RMSE | CP | ||||
0 | 1 | −3 | 1 | 0.053 | 1.140 | 1.141 | 0.903 | 0.031 | 0.882 | 0.885 | 0.909 | 0.013 | 0.662 | 0.675 | 0.912 | 0.009 | 0.449 | 0.458 | 0.939 | |
0.084 | 0.489 | 0.496 | 0.972 | 0.069 | 0.367 | 0.373 | 0.965 | 0.052 | 0.278 | 0.287 | 0.961 | 0.042 | 0.197 | 0.203 | 0.959 | |||||
−0.176 | 2.336 | 2.336 | 0.887 | −0.093 | 1.914 | 1.936 | 0.892 | −0.053 | 1.366 | 1.398 | 0.906 | −0.042 | 0.892 | 0.910 | 0.921 | |||||
q | 0.195 | 0.587 | 0.618 | 0.973 | 0.119 | 0.350 | 0.370 | 0.972 | 0.082 | 0.220 | 0.235 | 0.965 | 0.057 | 0.151 | 0.161 | 0.961 | ||||
0 | 1 | −1 | 1 | −0.099 | 1.473 | 1.473 | 0.918 | −0.078 | 1.045 | 1.049 | 0.928 | −0.051 | 0.704 | 0.711 | 0.935 | −0.033 | 0.473 | 0.477 | 0.948 | |
0.159 | 0.532 | 0.555 | 0.970 | 0.111 | 0.372 | 0.388 | 0.965 | 0.075 | 0.252 | 0.263 | 0.958 | 0.044 | 0.179 | 0.184 | 0.953 | |||||
−0.154 | 2.398 | 2.399 | 0.887 | −0.117 | 1.810 | 1.822 | 0.889 | −0.083 | 1.183 | 1.196 | 0.907 | −0.060 | 0.765 | 0.770 | 0.915 | |||||
q | 0.214 | 0.555 | 0.594 | 0.962 | 0.163 | 0.479 | 0.506 | 0.960 | 0.080 | 0.229 | 0.243 | 0.958 | 0.041 | 0.147 | 0.152 | 0.953 | ||||
0 | 1 | 3 | 1 | −0.100 | 1.166 | 1.170 | 0.901 | −0.091 | 0.907 | 0.913 | 0.928 | −0.077 | 0.615 | 0.624 | 0.936 | −0.047 | 0.448 | 0.458 | 0.945 | |
0.094 | 0.512 | 0.514 | 0.974 | 0.076 | 0.394 | 0.401 | 0.968 | 0.056 | 0.254 | 0.261 | 0.961 | 0.053 | 0.194 | 0.201 | 0.957 | |||||
0.251 | 2.300 | 2.299 | 0.871 | 0.192 | 1.843 | 1.875 | 0.896 | 0.167 | 1.295 | 1.322 | 0.917 | 0.094 | 0.864 | 0.887 | 0.925 | |||||
q | 0.168 | 0.657 | 0.678 | 0.965 | 0.115 | 0.376 | 0.393 | 0.961 | 0.071 | 0.220 | 0.231 | 0.960 | 0.045 | 0.148 | 0.154 | 0.959 | ||||
0 | 1 | −2 | 2 | −0.061 | 1.140 | 1.139 | 0.905 | −0.033 | 1.097 | 1.097 | 0.918 | −0.031 | 0.941 | 0.950 | 0.929 | −0.010 | 0.664 | 0.672 | 0.932 | |
0.053 | 0.386 | 0.389 | 0.986 | 0.042 | 0.343 | 0.357 | 0.979 | 0.034 | 0.297 | 0.318 | 0.971 | 0.025 | 0.189 | 0.200 | 0.964 | |||||
−0.192 | 2.875 | 2.880 | 0.894 | −0.151 | 2.778 | 2.783 | 0.901 | −0.122 | 2.318 | 2.355 | 0.915 | −0.097 | 1.625 | 1.652 | 0.921 | |||||
q | 0.362 | 1.301 | 1.350 | 0.971 | 0.278 | 1.254 | 1.341 | 0.969 | 0.226 | 1.008 | 1.094 | 0.963 | 0.180 | 0.510 | 0.541 | 0.959 | ||||
0 | 1 | 3 | 2 | −0.205 | 1.004 | 1.024 | 0.899 | −0.079 | 0.915 | 0.918 | 0.910 | −0.061 | 0.758 | 0.762 | 0.928 | −0.044 | 0.580 | 0.581 | 0.936 | |
0.079 | 0.389 | 0.389 | 0.971 | 0.053 | 0.313 | 0.314 | 0.968 | 0.041 | 0.264 | 0.271 | 0.961 | 0.030 | 0.201 | 0.205 | 0.959 | |||||
0.278 | 2.655 | 2.668 | 0.896 | 0.212 | 2.570 | 2.569 | 0.906 | 0.144 | 2.100 | 2.127 | 0.917 | 0.100 | 1.566 | 1.578 | 0.929 | |||||
q | 0.213 | 1.256 | 1.273 | 0.965 | 0.192 | 1.035 | 1.075 | 0.961 | 0.119 | 0.880 | 0.926 | 0.959 | 0.099 | 0.507 | 0.529 | 0.958 | ||||
0 | 1 | 2 | 2 | 0.164 | 1.171 | 1.191 | 0.904 | 0.115 | 1.114 | 1.115 | 0.912 | 0.079 | 0.901 | 0.904 | 0.939 | 0.052 | 0.689 | 0.693 | 0.945 | |
0.099 | 0.369 | 0.369 | 0.986 | 0.082 | 0.335 | 0.350 | 0.982 | 0.074 | 0.280 | 0.299 | 0.971 | 0.040 | 0.204 | 0.215 | 0.960 | |||||
0.381 | 3.081 | 3.103 | 0.866 | 0.242 | 2.772 | 2.782 | 0.899 | 0.171 | 2.252 | 2.267 | 0.919 | 0.127 | 1.668 | 1.683 | 0.935 | |||||
q | 0.280 | 1.318 | 1.347 | 0.976 | 0.223 | 1.137 | 1.213 | 0.973 | 0.112 | 1.023 | 1.103 | 0.967 | 0.087 | 0.583 | 0.629 | 0.959 | ||||
0 | 1 | −1 | 3 | 0.023 | 1.102 | 1.102 | 0.919 | 0.022 | 1.059 | 1.059 | 0.932 | 0.005 | 1.013 | 1.013 | 0.938 | 0.004 | 0.913 | 0.914 | 0.941 | |
0.102 | 0.321 | 0.321 | 0.979 | 0.071 | 0.246 | 0.256 | 0.975 | 0.068 | 0.222 | 0.251 | 0.971 | 0.057 | 0.191 | 0.224 | 0.961 | |||||
−0.134 | 3.189 | 3.188 | 0.972 | −0.116 | 3.001 | 3.002 | 0.965 | −0.072 | 2.726 | 2.725 | 0.961 | −0.049 | 2.414 | 2.417 | 0.955 | |||||
q | 0.386 | 1.835 | 1.837 | 0.974 | 0.320 | 1.526 | 1.582 | 0.970 | 0.202 | 1.420 | 1.484 | 0.965 | 0.170 | 1.258 | 1.244 | 0.961 | ||||
0 | 1 | 2 | 3 | 0.128 | 1.014 | 1.022 | 0.899 | 0.069 | 0.930 | 0.930 | 0.912 | 0.061 | 0.830 | 0.832 | 0.943 | 0.046 | 0.813 | 0.815 | 0.944 | |
0.134 | 0.324 | 0.325 | 0.984 | 0.079 | 0.274 | 0.278 | 0.977 | 0.052 | 0.246 | 0.266 | 0.962 | 0.049 | 0.209 | 0.227 | 0.959 | |||||
0.159 | 2.939 | 2.942 | 0.888 | 0.128 | 2.739 | 2.752 | 0.911 | 0.116 | 2.643 | 2.655 | 0.927 | 0.082 | 2.253 | 2.264 | 0.935 | |||||
q | 0.322 | 1.556 | 1.556 | 0.981 | 0.262 | 1.411 | 1.475 | 0.972 | 0.174 | 1.271 | 1.237 | 0.961 | 0.054 | 1.027 | 1.039 | 0.958 | ||||
−5 | 4 | −2 | 2 | −0.354 | 4.484 | 4.495 | 0.900 | −0.225 | 4.405 | 4.405 | 0.917 | −0.137 | 3.562 | 3.600 | 0.927 | −0.040 | 2.641 | 2.670 | 0.935 | |
0.317 | 1.550 | 1.553 | 0.970 | 0.281 | 1.311 | 1.365 | 0.964 | 0.237 | 1.070 | 1.132 | 0.958 | 0.170 | 0.793 | 0.837 | 0.957 | |||||
−0.247 | 2.862 | 2.861 | 0.885 | −0.223 | 2.808 | 2.817 | 0.906 | −0.191 | 2.240 | 2.273 | 0.919 | −0.130 | 1.631 | 1.658 | 0.929 | |||||
q | 0.309 | 1.292 | 1.328 | 0.974 | 0.245 | 1.155 | 1.237 | 0.966 | 0.222 | 0.851 | 0.909 | 0.961 | 0.157 | 0.581 | 0.617 | 0.955 | ||||
10 | 16 | 1 | 3 | 0.677 | 17.507 | 17.571 | 0.912 | 0.594 | 13.643 | 13.944 | 0.914 | 0.430 | 11.891 | 11.889 | 0.926 | 0.184 | 9.222 | 9.240 | 0.945 | |
1.237 | 4.948 | 4.951 | 0.904 | 0.957 | 4.159 | 4.205 | 0.927 | 0.651 | 3.454 | 3.827 | 0.937 | 0.480 | 2.030 | 2.113 | 0.949 | |||||
0.128 | 3.206 | 3.204 | 0.898 | 0.118 | 3.084 | 3.088 | 0.915 | 0.102 | 2.726 | 2.727 | 0.922 | 0.075 | 2.339 | 2.344 | 0.938 | |||||
q | 0.470 | 1.401 | 1.401 | 0.981 | 0.453 | 1.273 | 1.240 | 0.979 | 0.333 | 1.087 | 1.052 | 0.961 | 0.142 | 0.927 | 0.938 | 0.959 |
n | S | |||||
---|---|---|---|---|---|---|
76 |
Estimations | TESN | STESN |
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q | – | |
log-likelihood | ||
AIC | ||
BIC | ||
KSS | ||
p-value |
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Rivera, P.A.; Gallardo, D.I.; Venegas, O.; Bourguignon, M.; Gómez, H.W. An Extension of the Truncated-Exponential Skew- Normal Distribution. Mathematics 2021, 9, 1894. https://doi.org/10.3390/math9161894
Rivera PA, Gallardo DI, Venegas O, Bourguignon M, Gómez HW. An Extension of the Truncated-Exponential Skew- Normal Distribution. Mathematics. 2021; 9(16):1894. https://doi.org/10.3390/math9161894
Chicago/Turabian StyleRivera, Pilar A., Diego I. Gallardo, Osvaldo Venegas, Marcelo Bourguignon, and Héctor W. Gómez. 2021. "An Extension of the Truncated-Exponential Skew- Normal Distribution" Mathematics 9, no. 16: 1894. https://doi.org/10.3390/math9161894