Nonparametric Sieve Maximum Likelihood Estimation of Semi-Competing Risks Data
Abstract
:1. Introduction
2. Methodology
2.1. Model and Likelihood Function
2.2. Sieve Space for the Parameters
2.3. Maximization
3. Theoretical Properties
Assumptions
- (A1) and have compact supports (say ) and X has bounded support in where q is the dimension of Moreover, if there exists a constant and a constant vector such that almost surely, then and
- (A2) where is a compact set of with nonempty interior. and and
- (A3) where satisfies the restrictions
- (A4) where r is the measure of smoothness of in definitions of and
- (A5) for any is continuously differentiable in near and
4. Simulation Study
5. A Real Data Example
6. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Proofs of Theorem 1, Theorem 2, and Theorem 3
Appendix A.1. Proof of Theorem 1
Appendix A.2. Proof of Theorem 2
Appendix A.3. Proof of Theorem 3
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BIAS | STD | ESE | CP | ||
---|---|---|---|---|---|
0.021 | 0.233 | 0.219 | 0.953 | ||
−0.016 | 0.230 | 0.263 | 0.954 | ||
0.026 | 0.281 | 0.219 | 0.986 | ||
0.017 | 0.166 | 0.159 | 0.963 | ||
−0.013 | 0.167 | 0.164 | 0.960 | ||
0.018 | 0.122 | 0.141 | 0.965 |
BIAS | STD | ESE | CP | ||
---|---|---|---|---|---|
= −1 | −0.015 | 0.244 | 0.225 | 0.956 | |
= −1 | 0.019 | 0.232 | 0.239 | 0.962 | |
= −1 | −0.014 | 0.269 | 0.284 | 0.961 | |
= −1 | −0.013 | 0.144 | 0.165 | 0.961 | |
= −1 | 0.014 | 0.158 | 0.164 | 0.945 | |
= −1 | −0.013 | 0.197 | 0.185 | 0.980 |
BIAS | STD | ESE | CP | ||
---|---|---|---|---|---|
0.017 | 0.230 | 0.205 | 0.966 | ||
−0.013 | 0.221 | 0.219 | 0.965 | ||
0.016 | 0.182 | 0.218 | 0.945 | ||
0.008 | 0.172 | 0.155 | 0.941 | ||
−0.011 | 0.132 | 0.152 | 0.954 | ||
0.012 | 0.125 | 0.157 | 0.938 |
Transition | Parameters | Estimate | Standard Error | p-Value |
---|---|---|---|---|
12 | −0.513 | 0.119 | 1.6 × 10 | |
13 | −0.028 | 0.379 | 0.469 | |
23 | 0.738 | 0.130 | 7.0 × 10 |
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Huang, X.; Xu, J. Nonparametric Sieve Maximum Likelihood Estimation of Semi-Competing Risks Data. Mathematics 2022, 10, 2248. https://doi.org/10.3390/math10132248
Huang X, Xu J. Nonparametric Sieve Maximum Likelihood Estimation of Semi-Competing Risks Data. Mathematics. 2022; 10(13):2248. https://doi.org/10.3390/math10132248
Chicago/Turabian StyleHuang, Xifen, and Jinfeng Xu. 2022. "Nonparametric Sieve Maximum Likelihood Estimation of Semi-Competing Risks Data" Mathematics 10, no. 13: 2248. https://doi.org/10.3390/math10132248
APA StyleHuang, X., & Xu, J. (2022). Nonparametric Sieve Maximum Likelihood Estimation of Semi-Competing Risks Data. Mathematics, 10(13), 2248. https://doi.org/10.3390/math10132248