Title Page
Abstract
Contents
Chapter 1. Introduction 18
Chapter 2. Epistemic confidence in the observed confidence interval 23
2.1. Introduction 23
2.2. Main theory 27
2.2.1. Relevant subsets 27
2.2.2. Confidence distribution 29
2.2.3. Main theorem 32
2.2.4. Ancillary statistics 37
2.2.5. Computation of epistemic confidence 40
2.3. Examples 41
2.3.1. Simple model 41
2.3.2. Location family model 42
2.3.3. Exponential family model 43
2.4. Discussion 52
2.5. Appendix 54
2.5.1. Curved exponential model: N(θ, θ²) 54
2.5.2. Discrete case 58
2.5.3. When maximal ancillaries are not unique 61
Chapter 3. Point mass in confidence distributions 64
3.1. Introduction 64
3.2. Ambiguity in confidence of an observed CI 67
3.3. GFD and probability dilution 71
3.4. CD and related methods 74
3.4.1. CD and confidence level of an observed CI 76
3.4.2. CD versus GFD 79
3.4.3. CD versus RP 81
3.5. On false confidence 83
3.5.1. False confidence and probability dilution 86
3.6. Hypothesis testing 88
3.7. Concluding remarks 91
3.8. Appendix 92
3.8.1. Proof of Theorem 3.1 92
Chapter 4. Foundations of h-likelihood inference for random unknowns 94
4.1. Introduction 94
4.2. Hierarchical likelihood 96
4.2.1. Reformulation of H-Likelihood 99
4.2.2. Bartlizable scale of random effects 103
4.3. Main Results 108
4.4. H-likelihood theory for irregular cases 112
4.4.1. Missing data problem when ˆv − v =Op(1)[이미지참조] 113
4.4.2. Exponential-exponential HGLM when ˆθ−θ =Op(1)[이미지참조] 119
4.5. When the h-likelihood is not explicit 124
4.6. Appendix 125
Chapter 5. DNN with temporal-spatial random effects via h-likelihood 136
5.1. Introduction 136
5.2. Integrated likelihood approach for LMMs 138
5.3. H-likelihood approach for LMMs 141
5.4. Learning algorithm with the h-likelihood 145
5.4.1. REML procedure 149
5.4.2. Adjustments for random effects 150
5.5. Comparison with existing methods 151
5.6. Numerical studies 153
5.7. Real data analysis 156
5.8. Concluding remarks 158
5.9. Appendix 160
5.9.1. The computation of h-likelihood when q₁ is large 160
5.9.2. Methods-of-moments estimators 161
5.9.3. Technical details 161
Chapter 6. DNN for clustered count data via h-likelihood 167
6.1. Introduction 167
6.2. Model Descriptions 169
6.2.1. Poisson DNN 169
6.2.2. Poisson-gamma DNN 170
6.3. Construction of h-likelihood 172
6.4. Learning algorithm with the h-likelihood 175
6.4.1. Loss function for online learning 175
6.4.2. The local minima problem 176
6.4.3. Pretraining variance components 177
6.5. Experimental Studies 178
6.6. Real Data Analysis 183
6.7. Concluding Remarks 185
6.8. Appendix 185
6.8.1. Convergence of the method-of-moments estimator 185
6.8.2. Technical details 187
Chapter 7. DNN for semi-parametric frailty models via h-likelihood 190
7.1. Introduction 190
7.2. A review of DNN-Cox model 191
7.2.1. DNN-Cox model 191
7.2.2. Prediction measures 193
7.3. Proposed DNN for frailty model 195
7.3.1. DNN-FM 196
7.3.2. Construction of h-likelihood 196
7.4. Learning algorithm using the profiled h-likelihood 199
7.4.1. Local minima problem 201
7.4.2. ML learning algorithm 201
7.5. Experimental studies 203
7.5.1. Experimental design 203
7.5.2. Experimental results 205
7.6. Multi-center bladder cancer data 211
7.7. Concluding remarks 214
7.8. Appendix 215
7.8.1. Derivation for the predictive likelihood 215
7.8.2. Evaluation measures for DNN-FM 216
7.8.3. Online learning for the DNN-FM 218
7.8.4. Proofs 220
Bibliography 225
국문초록 241
Table 5.1. Average of test MSPEs. Results of existing models are cited from Simchoni and Rosset (2023). 155
Table 5.2. Estimated variance components on average when σe²=vσ²=1. Results of LMMNN-R are cited from Simchoni and Rosset (2023). 155
Table 5.3. Average of test MSPEs from the 5-fold cross validation. Results of existing models are cited from Simchoni and Rosset (2023). 158
Table 6.1. Averages of test RMSEs and RMSRs of simulation studies over 100 replications of each scenario. G(0) implies the absence of random effects, i.e.,... 182
Table 6.2. Test RMSEs and RMSRs of real data analyses. Bold numbers are the minimum values. 186
Table 7.1. Mean (standard deviation) of IBS and C-index from 100 replications for each frailty variance α. 210
Table 7.2. Mean and standard deviation of estimated frailty variance bα from 100 replications. 211
Table 7.3. IBS and C-index for four survival prediction models on the test set of the bladder cancer data 214
Figure 2.1. (a) Implied priors of θ computed from (2.20) (solid) and (2.19) (◦). Both are normalized such that they are equal to one at the MLE (•). (b) The... 46
Figure 2.2. cfi(θ) for i=1, 2, 3 and cm(θ) (circle) based on (a) y=(0.9, 1, 1.5) and (b) y=(0.1, 1, 5). In each panel, all three curves cfi are actually drawn in...[이미지참조] 48
Figure 2.3. Approximations of the conditional P-value Pθ(T ≥ t|a) to test H0: θ=1 from N(θ, θ²). The x-axis is the exact theoretical value based on...[이미지참조] 51
Figure 2.4. cfi(θ; ˆθi) for i=1, ..., n (solid), cf(θ; y) (circle) and cm(θ; ˆθ₁,···, ˆθn) (cross). (a) n=3, (b) n=10.[이미지참조] 56
Figure 2.5. Coverage probabilities of the intervals from the confidence distribution based on the mid p-value for binomial models at n=10, 50, 100 (top)... 60
Figure 3.1. Confidence intervals with (a) α=0.95, (b) α=0.90, (c) α=0.60. For all three CIs, β=0.05. 72
Figure 3.2. Average of C(θ; d) and G(θ, d) over 10,000 repeats. 80
Figure 3.3. Coverage probabilities of 80% CIs based on CD, GFD, and RP when k=2 and k=100. 82
Figure 3.4. Average confidences and beliefs regarding collision over 100,000 repetitions. 87
Figure 4.1. The h-likelihood (blue), the marginal likelihood (red), and the profile likelihood (orange) are computed with N=2000 samples from yi ~...[이미지참조] 106
Figure 5.1. A sketch of the proposed model fitting algorithm via h-likelihood. 145
Figure 5.2. MSE curves of the integrated likelihood approach and the proposed h-likelihood approach from 20 repetitions. N=10, 000 data are generated... 149
Figure 5.3. The HL predictors from income data (left) and air quality data (right). 156
Figure 6.1. Predicted values of ui from two replications (marked as o and x for each) when ui is generated from the Gamma distribution with λ=1, n=100,... 177
Figure 6.2. Learning curve for the variance component λ when (a) λ=0, (b) λ=0.5, and (c) λ=1. 178
Figure 7.1. An example of model architecture for DNN-FM. 197
Figure 7.2. A schematic diagram of DNN-FM fitting procedure. 202
Figure 7.3. 15% censoring: Box plot of IBS from 100 replications for each frailty variance, var(u)=α. 206
Figure 7.4. 15% censoring: Box plot of C-index from 100 replications for each frailty variance, var(u)=α. 207
Figure 7.5. 45% censoring: Box plot of IBS from 100 replications for each frailty variance, var(u)=α. 208
Figure 7.6. 45% censoring: Box plot of C-index from 100 replications for each frailty variance, var(u)=α. 209
Figure 7.7. Time-dependent Brier score for four survival prediction models on the test set of the bladder cancer data. 213