Estimation and Model Misspecification for Recurrent Event Data with Covariates Under Measurement Errors
Abstract
1. Introduction
2. Preliminaries
2.1. Dynamic Modeling and Observables
2.2. Measurement Errors Notations
3. Corrected Scores and Estimators
3.1. Preliminary
- (i)
- likelihood-based, where a specification of the distribution of the covariates is usually conjectured;
- (ii)
- methods based on unbiased estimating functions;
- (iii)
- methods based on correcting a naive parameter estimate, i.e., the one obtained ignoring the errors.
3.2. Large Sample Properties Under Correction
3.3. Measurement Errors Variance Estimation
4. Simulation and Application
4.1. Simulation Results
4.2. Application: rhDNase Data
5. Corrected Baseline Hazard and Properties
6. Misspecified Errors Model
Properties of Corrected Estimator Under Misspecified Errors
- (a)
- (b)
- where is given by
7. Conclusions and Discussion
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Technical Appendix
- The s are independent with mean zero and independent of .
- and is time-independent.
- The moment-generating functions at all orders exist.
- .
- is uniformly bounded for for all i.
- For and 2 there exist functions and a neighborhood of such that the functions are continuous functions of uniformly in and bounded on Furthermore,
- Let be as in Condition III. For all and , define
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−1.0 | −0.839 | 0.095 | −0.980 | 0.143 | 0.978 | 0.238 | 1.036 | 0.240 | ||
−0.8 | −0.678 | 0.109 | −0.836 | 0.158 | 1.039 | 0.214 | 1.112 | 0.217 | ||
−0.4 | −0.365 | 0.117 | −0.415 | 0.106 | 1.054 | 0.238 | 1.049 | 0.240 | ||
−0.2 | −0.177 | 0.117 | −0.230 | 0.128 | 1.062 | 0.249 | 1.040 | 0.247 | ||
40 | −0.1 | −0.078 | 0.080 | −0.081 | 0.120 | 1 | 1.002 | 0.248 | 1.040 | 0.230 |
0.1 | 0.096 | 0.085 | 0.093 | 0.124 | 1.052 | 0.238 | 1.063 | 0.193 | ||
0.2 | 0.170 | 0.079 | 0.183 | 0.125 | 1.019 | 0.260 | 1.028 | 0.207 | ||
0.4 | 0.347 | 0.101 | 0.411 | 0.134 | 1.013 | 0.208 | 1.024 | 0.256 | ||
0.8 | 0.709 | 0.138 | 0.773 | 0.150 | 1.067 | 0.260 | 1.079 | 0.244 | ||
1.0 | 0.859 | 0.116 | 1.021 | 0.169 | 0.968 | 0.196 | 1.111 | 0.270 | ||
−1.0 | −0.840 | 0.079 | −0.947 | 0.093 | 0.991 | 0.153 | 1.008 | 0.170 | ||
−0.8 | −0.687 | 0.079 | −0.801 | 0.088 | 1.018 | 0.166 | 1.052 | 0.187 | ||
−0.4 | −0.342 | 0.064 | −0.395 | 0.084 | 1.005 | 0.171 | 1.030 | 0.156 | ||
−0.2 | −0.187 | 0.062 | −0.187 | 0.079 | 1.035 | 0.165 | 1.031 | 0.158 | ||
80 | −0.1 | −0.083 | 0.068 | −0.105 | 0.059 | 1 | 1.019 | 0.164 | 0.976 | 0.154 |
0.1 | 0.075 | 0.062 | 0.090 | 0.078 | 1.026 | 0.177 | 1.051 | 0.131 | ||
0.2 | 0.195 | 0.062 | 0.187 | 0.056 | 1.035 | 0.128 | 1.024 | 0.160 | ||
0.4 | 0.338 | 0.064 | 0.385 | 0.089 | 1.006 | 0.139 | 0.980 | 0.139 | ||
0.8 | 0.665 | 0.075 | 0.784 | 0.093 | 0.972 | 0.156 | 1.046 | 0.146 | ||
1.0 | 0.831 | 0.070 | 0.959 | 0.100 | 1.002 | 0.137 | 1.029 | 0.158 | ||
−1.0 | −0.826 | 0.069 | −0.984 | 0.100 | 0.960 | 0.132 | 1.058 | 0.140 | ||
−0.8 | −0.674 | 0.060 | −0.807 | 0.086 | 0.984 | 0.127 | 1.017 | 0.127 | ||
−0.4 | −0.354 | 0.063 | −0.388 | 0.066 | 1.040 | 0.151 | 1.007 | 0.119 | ||
−0.2 | −0.177 | 0.060 | −0.199 | 0.068 | 1.031 | 0.146 | 1.006 | 0.135 | ||
−0.1 | −0.086 | 0.064 | −0.101 | 0.058 | 0.993 | 0.126 | 1.033 | 0.137 | ||
100 | 0.1 | 0.093 | 0.059 | 0.103 | 0.056 | 1 | 1.028 | 0.142 | 0.988 | 0.131 |
0.2 | 0.183 | 0.063 | 0.211 | 0.072 | 1.057 | 0.140 | 1.029 | 0.115 | ||
0.4 | 0.367 | 0.063 | 0.399 | 0.070 | 1.016 | 0.144 | 1.017 | 0.115 | ||
0.8 | 0.685 | 0.070 | 0.784 | 0.083 | 1.011 | 0.141 | 1.026 | 0.166 | ||
1.0 | 0.839 | 0.068 | 0.965 | 0.094 | 0.989 | 0.141 | 1.010 | 0.149 | ||
−1.0 | −0.821 | 0.039 | −0.962 | 0.065 | 1.018 | 0.089 | 1.004 | 0.106 | ||
−0.8 | −0.668 | 0.053 | −0.763 | 0.051 | 0.969 | 0.097 | 1.026 | 0.103 | ||
−0.4 | −0.338 | 0.039 | −0.401 | 0.051 | 1.011 | 0.112 | 1.022 | 0.103 | ||
−0.2 | −0.182 | 0.046 | −0.210 | 0.049 | 1.028 | 0.105 | 1.014 | 0.093 | ||
−0.1 | −0.092 | 0.045 | −0.109 | 0.044 | 1.015 | 0.081 | 1.012 | 0.093 | ||
200 | 0.1 | 0.079 | 0.035 | 0.101 | 0.043 | 1 | 1.013 | 0.081 | 1.048 | 0.109 |
0.2 | 0.177 | 0.042 | 0.208 | 0.047 | 1.016 | 0.099 | 1.022 | 0.089 | ||
0.4 | 0.348 | 0.046 | 0.404 | 0.040 | 1.004 | 0.095 | 1.033 | 0.100 | ||
0.8 | 0.678 | 0.043 | 0.758 | 0.050 | 0.976 | 0.107 | 1.001 | 0.099 | ||
1.0 | 0.832 | 0.057 | 0.968 | 0.067 | 1.015 | 0.101 | 1.036 | 0.111 |
n | 40 | 80 | 100 | 200 |
---|---|---|---|---|
MSE(naive) | 0.0016 | 0.0005 | 0.0009 | 0.0004 |
MSE(corrected) | 0.0041 | 0.0011 | 0.0007 | 0.0006 |
0.5 | −0.704 | 0.085 | −1.024 | 0.228 | 0.988 | 0.231 | 1.046 | 0.295 | |
0.4 | −0.737 | 0.079 | −0.980 | 0.193 | 1.012 | 0.224 | 1.129 | 0.265 | |
0.3 | −0.848 | 0.094 | −0.965 | 0.155 | 0.960 | 0.231 | 1.042 | 0.256 | |
40 | 0.2 | −0.909 | 0.107 | −0.979 | 0.135 | 0.979 | 0.237 | 1.055 | 0.243 |
0.1 | −0.926 | 0.099 | −0.911 | 0.116 | 1.029 | 0.227 | 1.036 | 0.193 | |
0.05 | −0.947 | 0.122 | −0.976 | 0.120 | 1.018 | 0.211 | 1.099 | 0.234 | |
0.01 | −0.938 | 0.144 | −0.928 | 0.134 | 0.956 | 0.177 | 1.044 | 0.227 | |
0.5 | −0.685 | 0.071 | −0.983 | 0.144 | 0.916 | 0.155 | 1.055 | 0.181 | |
0.4 | −0.772 | 0.086 | −0.929 | 0.086 | 0.966 | 0.152 | 1.049 | 0.162 | |
0.3 | −0.840 | 0.077 | −0.981 | 0.115 | 0.960 | 0.126 | 1.003 | 0.130 | |
80 | 0.2 | −0.930 | 0.091 | −0.983 | 0.089 | 1.005 | 0.128 | 1.025 | 0.150 |
0.1 | −0.911 | 0.066 | −0.940 | 0.082 | 0.989 | 0.135 | 1.056 | 0.138 | |
0.05 | −0.957 | 0.081 | −0.971 | 0.080 | 1.068 | 0.124 | 1.014 | 0.157 | |
0.01 | −0.941 | 0.089 | −0.933 | 0.071 | 1.005 | 0.132 | 1.001 | 0.161 | |
0.5 | −0.683 | 0.068 | −0.957 | 0.139 | 0.956 | 0.133 | 1.015 | 0.173 | |
0.4 | −0.773 | 0.056 | −0.975 | 0.108 | 1.026 | 0.143 | 1.118 | 0.173 | |
0.3 | −0.842 | 0.075 | −0.976 | 0.090 | 0.971 | 0.127 | 1.058 | 0.133 | |
100 | 0.2 | −0.937 | 0.074 | −1.002 | 0.091 | 1.011 | 0.135 | 0.996 | 0.136 |
0.1 | −0.936 | 0.063 | −0.957 | 0.078 | 0.997 | 0.137 | 1.042 | 0.124 | |
0.05 | −0.966 | 0.066 | −0.939 | 0.075 | 1.016 | 0.129 | 1.029 | 0.133 | |
0.01 | −0.936 | 0.068 | −0.947 | 0.071 | 1.011 | 0.154 | 1.024 | 0.120 | |
0.5 | −0.695 | 0.045 | −0.938 | 0.065 | 0.950 | 0.084 | 1.022 | 0.104 | |
0.4 | −0.766 | 0.042 | −0.941 | 0.059 | 1.000 | 0.081 | 1.095 | 0.110 | |
0.3 | −0.833 | 0.052 | −0.976 | 0.057 | 0.995 | 0.084 | 1.021 | 0.085 | |
200 | 0.2 | −0.929 | 0.051 | −0.977 | 0.059 | 1.004 | 0.108 | 1.029 | 0.097 |
0.1 | −0.915 | 0.044 | −0.938 | 0.047 | 0.978 | 0.076 | 1.004 | 0.094 | |
0.05 | −0.944 | 0.055 | −0.945 | 0.054 | 1.027 | 0.098 | 1.037 | 0.073 | |
0.01 | −0.937 | 0.051 | −0.932 | 0.049 | 1.013 | 0.109 | 1.010 | 0.087 |
0.000 | −1.7025 | −0.2734 | −1.7004 | −0.2729 |
0.050 | −1.6201 | −0.2695 | −1.6884 | −0.2678 |
0.100 | −1.4338 | −0.2665 | −1.6763 | −0.2644 |
0.150 | −1.2153 | −0.2646 | −1.6612 | −0.2603 |
0.200 | −1.0102 | −0.2636 | −1.6446 | −0.2566 |
0.300 | −0.6944 | −0.2633 | −1.6125 | −0.2467 |
0.400 | −0.4910 | −0.2640 | −1.5873 | −0.2370 |
0.500 | −0.3610 | −0.2649 | −1.5774 | −0.2240 |
0.600 | −0.2752 | −0.2657 | −1.6094 | −0.2438 |
0.700 | −0.2166 | −0.2665 | 421.8588 | 86.7149 |
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Alahakoon, R.; Zamba, G.K.D.; Wen, X.M.; Adekpedjou, A. Estimation and Model Misspecification for Recurrent Event Data with Covariates Under Measurement Errors. Mathematics 2025, 13, 113. https://doi.org/10.3390/math13010113
Alahakoon R, Zamba GKD, Wen XM, Adekpedjou A. Estimation and Model Misspecification for Recurrent Event Data with Covariates Under Measurement Errors. Mathematics. 2025; 13(1):113. https://doi.org/10.3390/math13010113
Chicago/Turabian StyleAlahakoon, Ravinath, Gideon K. D. Zamba, Xuerong Meggie Wen, and Akim Adekpedjou. 2025. "Estimation and Model Misspecification for Recurrent Event Data with Covariates Under Measurement Errors" Mathematics 13, no. 1: 113. https://doi.org/10.3390/math13010113
APA StyleAlahakoon, R., Zamba, G. K. D., Wen, X. M., & Adekpedjou, A. (2025). Estimation and Model Misspecification for Recurrent Event Data with Covariates Under Measurement Errors. Mathematics, 13(1), 113. https://doi.org/10.3390/math13010113