Identification and Empirical Likelihood Inference in Nonlinear Regression Model with Nonignorable Nonresponse
Abstract
:1. Introduction
2. Methods
2.1. Penalized Semiparametric Likelihood Estimation
2.2. Construction of Estimating Equations
2.3. MELEs of Model Parameters
3. Main Results
3.1. Asymptotic Properties
- (A1)
- The nonresponse mechanism almost surely and almost surely; in a neighborhood of , , and exists and is bounded by an integrable function.
- (A2)
- The probability density function is bounded away from ∞ in the support of ; the first and second derivatives of are continuous, smooth and bounded; and and are finite.
- (A3)
- is twice continuously differentiable in the neighborhood of .
- (A4)
- The function is continuous with respect to , where lies in a compact set; and exist; has full column rank.
- (A5)
- has full column rank.
- (A6)
- The kernel function is a probability density function such that (a) it is bounded and has a compact support; (b) it is symmetric with ; (c) for some in some closed interval centered at zero; and (d) the bandwidth h satisfies and as .
- (A7)
- As , , and the tuning parameter satisfies as and .
- (A8)
- The penalty function satisfies and , where .
- (A9)
- The moment conditions
- (1)
- Asymptotic normality:
- (2)
- Likelihood ratio convergence:
3.2. Double Robustness
3.3. Dimension Reduction
3.4. Asymptotic Variance Estimation
4. Simulation Study
4.1. Simulation 1
4.2. Simulation 2
5. Application to the ACTG 175 Data
- Treatment assignment (: 0 = ZDV monotherapy)
- Baseline CD4 count (: )
- Demographic covariates: age (), weight (), race (: 0 = White), gender (: 0 = Female)
- Clinical covariates: antiretroviral history (: 0 = naive), early treatment termination (: 0 = completed)
6. Conclusions and Future Work
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
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Est. | Bias | SD | RMS | T | F | Bias | SD | RMS | T | F |
---|---|---|---|---|---|---|---|---|---|---|
0.0549 | 0.1021 | 0.1158 | 3.69 | 0 | 0.0447 | 0.0703 | 0.0833 | 3.79 | 0 | |
0.0211 | 0.2313 | 0.2321 | – | – | 0.0038 | 0.1612 | 0.1612 | – | – | |
0.0031 | 0.1397 | 0.1397 | – | – | 0.0118 | 0.0984 | 0.0990 | – | – |
IPW | AIPW | ||||||
---|---|---|---|---|---|---|---|
Est. | Bias | SD | RMS | Bias | SD | RMS | |
150 | 0.0014 | 0.0442 | 0.0443 | 0.0007 | 0.0439 | 0.0439 | |
0.0015 | 0.0519 | 0.0520 | 0.0012 | 0.0520 | 0.0522 | ||
0.0015 | 0.0569 | 0.0570 | 0.0008 | 0.0576 | 0.0576 | ||
0.0006 | 0.0596 | 0.0596 | 0.0001 | 0.0601 | 0.0601 | ||
0.0023 | 0.0573 | 0.0574 | 0.0020 | 0.0574 | 0.0574 | ||
0.0011 | 0.0305 | 0.0305 | 0.0009 | 0.0305 | 0.0305 | ||
250 | 0.0006 | 0.0382 | 0.0382 | 0.0008 | 0.0379 | 0.0379 | |
0.0012 | 0.0399 | 0.0399 | 0.0012 | 0.0401 | 0.0402 | ||
0.0015 | 0.0466 | 0.0466 | 0.0013 | 0.0466 | 0.0466 | ||
0.0010 | 0.0450 | 0.0450 | 0.0013 | 0.0449 | 0.0449 | ||
0.0016 | 0.0409 | 0.0409 | 0.0021 | 0.0409 | 0.0409 | ||
0.0005 | 0.0227 | 0.0227 | 0.0007 | 0.0225 | 0.0225 |
Est. | Estimate | p-Value | Est. | Estimate | p-Value |
---|---|---|---|---|---|
0.64 | <0.001 | 0.0068 | <0.001 | ||
−0.0007 | <0.001 | 0.0002 | 0.002 | ||
0.0011 | 0.003 | 0.0010 | <0.001 | ||
0 | 0.574 | −0.6299 | <0.001 | ||
0 | 0.191 | −0.0010 | <0.001 |
Complete-Case Analysis | Han’s Method | |||||
---|---|---|---|---|---|---|
Estimate | s.e. | p-Value | Estimate | s.e. | p-Value | |
Intercept | 21.50 | 27.44 | 0.433 | 65.53 | 34.06 | 0.054 |
Trt | 63.68 | 9.09 | <0.001 | 52.72 | 10.34 | <0.001 |
0.76 | 0.04 | <0.001 | 0.73 | 0.05 | <0.001 | |
Age | 0.10 | 0.45 | 0.816 | 0.14 | 0.55 | 0.796 |
Weight | 0.54 | 0.28 | 0.054 | 0.27 | 0.33 | 0.417 |
Race | −20.60 | 8.51 | 0.015 | −18.30 | 9.66 | 0.058 |
Gender | −10.73 | 10.79 | 0.320 | −16.54 | 11.34 | 0.145 |
History | −42.02 | 7.62 | <0.001 | −41.45 | 8.65 | <0.001 |
Offtrt | −80.72 | 9.62 | <0.001 | −86.87 | 10.31 | <0.001 |
IPW | AIPW | |||||
Estimate | s.e. | p-Value | Estimate | s.e. | p-Value | |
Intercept | 33.15 | 30.66 | 0.2796 | 34.28 | 30.77 | 0.2651 |
Trt | 62.14 | 9.52 | <0.001 | 61.77 | 9.60 | <0.001 |
0.76 | 0.05 | <0.001 | 0.76 | 0.05 | <0.001 | |
Age | 0.18 | 0.53 | 0.7278 | 0.18 | 0.54 | 0.7326 |
Weight | 0.42 | 0.32 | 0.1797 | 0.42 | 0.32 | 0.1884 |
Race | −22.07 | 10.10 | 0.0288 | −22.01 | 10.13 | 0.0297 |
Gender | −9.38 | 12.07 | 0.4369 | −9.33 | 12.10 | 0.4402 |
History | −41.34 | 8.67 | <0.001 | −41.24 | 8.70 | <0.001 |
Offtrt | −74.74 | 11.62 | <0.001 | −74.44 | 11.64 | <0.001 |
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Ding, X.; Li, X. Identification and Empirical Likelihood Inference in Nonlinear Regression Model with Nonignorable Nonresponse. Mathematics 2025, 13, 1388. https://doi.org/10.3390/math13091388
Ding X, Li X. Identification and Empirical Likelihood Inference in Nonlinear Regression Model with Nonignorable Nonresponse. Mathematics. 2025; 13(9):1388. https://doi.org/10.3390/math13091388
Chicago/Turabian StyleDing, Xianwen, and Xiaoxia Li. 2025. "Identification and Empirical Likelihood Inference in Nonlinear Regression Model with Nonignorable Nonresponse" Mathematics 13, no. 9: 1388. https://doi.org/10.3390/math13091388
APA StyleDing, X., & Li, X. (2025). Identification and Empirical Likelihood Inference in Nonlinear Regression Model with Nonignorable Nonresponse. Mathematics, 13(9), 1388. https://doi.org/10.3390/math13091388