Empirical Likelihood for Composite Quantile Regression Models with Missing Response Data
Abstract
:1. Introduction
2. Empirical Likelihood for Composite Quantile Regression with Missing Response Data
2.1. Complete-Case Linear Composite Quantile Regression Empirical Likelihood
2.2. Weighted Composite Quantile Empirical Likelihood
2.3. Imputation Composite Quantile Empirical Likelihood
3. Asymptotic Properties
4. Simulation Study
5. A Real-World Example
6. Conclusions and Discussions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
References
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QEL | NA | |||||
---|---|---|---|---|---|---|
IQEL | ICQEL | NA () | NA () | NCQEL | ||
100 | 0.9171 | 0.9199 | 0.9182 | 0.9012 | 0.9201 | |
150 | 0.9285 | 0.9298 | 0.9281 | 0.9121 | 0.9299 | |
200 | 0.9346 | 0.9379 | 0.9354 | 0.9243 | 0.9379 | |
100 | 0.9025 | 0.9046 | 0.9024 | 0.8998 | 0.9047 | |
150 | 0.9142 | 0.9158 | 0.9136 | 0.9032 | 0.9157 | |
200 | 0.9238 | 0.9298 | 0.9236 | 0.9198 | 0.9299 | |
100 | 0.9138 | 0.9198 | 0.9128 | 0.9016 | 0.9196 | |
150 | 0.9279 | 0.9312 | 0.9268 | 0.9189 | 0.9314 | |
200 | 0.9354 | 0.9416 | 0.9339 | 0.9296 | 0.9418 |
QEL | NA | |||||
---|---|---|---|---|---|---|
IQEL | ICQEL | NA () | NA () | NCQEL | ||
100 | 0.2891 | 0.2942 | 0.2812 | 0.2898 | 0.2940 | |
150 | 0.2678 | 0.2698 | 0.2659 | 0.2645 | 0.2699 | |
200 | 0.2264 | 0.2314 | 0.2245 | 0.2214 | 0.2315 | |
100 | 0.3452 | 0.3246 | 0.3358 | 0.3389 | 0.3244 | |
150 | 0.3345 | 0.3187 | 0.3298 | 0.3301 | 0.3186 | |
200 | 0.3254 | 0.2978 | 0.3187 | 0.3198 | 0.2975 | |
100 | 0.3165 | 0.3056 | 0.3127 | 0.3157 | 0.3056 | |
150 | 0.3106 | 0.2986 | 0.3097 | 0.3102 | 0.2985 | |
200 | 0.2997 | 0.2856 | 0.2898 | 0.2984 | 0.2855 |
QEL | NA | |||||
---|---|---|---|---|---|---|
IQEL | ICQEL | NA () | NA () | NCQEL | ||
100 | 0.9113 | 0.9251 | 0.9103 | 0.9098 | 0.9252 | |
150 | 0.9241 | 0.9312 | 0.9298 | 0.9214 | 0.9312 | |
200 | 0.9302 | 0.9389 | 0.9315 | 0.9299 | 0.9388 | |
100 | 0.9122 | 0.9298 | 0.9288 | 0.9119 | 0.9299 | |
150 | 0.9214 | 0.9384 | 0.9381 | 0.9207 | 0.9385 | |
200 | 0.9306 | 0.9476 | 0.9451 | 0.9302 | 0.9477 | |
100 | 0.9244 | 0.9285 | 0.9242 | 0.9231 | 0.9288 | |
150 | 0.9349 | 0.9364 | 0.9348 | 0.9316 | 0.9364 | |
200 | 0.9403 | 0.9478 | 0.9405 | 0.9399 | 0.9479 |
QEL | NA | |||||
---|---|---|---|---|---|---|
IQEL | ICQEL | NA () | NA () | NCQEL | ||
100 | 0.2879 | 0.2912 | 0.2876 | 0.2896 | 0.2911 | |
150 | 0.2798 | 0.2822 | 0.2788 | 0.2799 | 0.2822 | |
200 | 0.2614 | 0.2698 | 0.2612 | 0.2616 | 0.2697 | |
100 | 0.3124 | 0.3056 | 0.3087 | 0.3134 | 0.3055 | |
150 | 0.3045 | 0.3002 | 0.3012 | 0.3055 | 0.3002 | |
200 | 0.2978 | 0.2945 | 0.2968 | 0.2998 | 0.2944 | |
100 | 0.3015 | 0.3002 | 0.3011 | 0.3014 | 0.3003 | |
150 | 0.2978 | 0.2968 | 0.2971 | 0.2979 | 0.2967 | |
200 | 0.2868 | 0.2854 | 0.2861 | 0.2867 | 0.2853 |
Estimators | Confidence Intervals | |||||
---|---|---|---|---|---|---|
IQEL | ICQEL | NCQEL | IQEL | ICQEL | NCQEL | |
101.01 | 101.02 | 101.04 | (74.60, 112.21) | (76.25, 106.46) | (76.46, 106.51) | |
0.4993 | 0.4996 | 0.4995 | (0.4656, 0.5875) | (0.4724, 0.5648) | (0.4727, 0.5649) |
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Luo, S.; Zheng, Y.; Zhang, C.-y. Empirical Likelihood for Composite Quantile Regression Models with Missing Response Data. Symmetry 2024, 16, 1314. https://doi.org/10.3390/sym16101314
Luo S, Zheng Y, Zhang C-y. Empirical Likelihood for Composite Quantile Regression Models with Missing Response Data. Symmetry. 2024; 16(10):1314. https://doi.org/10.3390/sym16101314
Chicago/Turabian StyleLuo, Shuanghua, Yu Zheng, and Cheng-yi Zhang. 2024. "Empirical Likelihood for Composite Quantile Regression Models with Missing Response Data" Symmetry 16, no. 10: 1314. https://doi.org/10.3390/sym16101314