臺灣博碩士論文加值系統

線性轉換模型為一個相當彈性的半參數迴歸模型。在存活分析中，最常被使用在分析上的比例風險模型以及比例勝算模型皆為線性轉換模型的兩個特例。因此，本篇研究將探討在線性轉換模型之下，利用核密度估計方法對未知累加基底風險函數進行估計的表現。本篇選用Nadaraya-Watson核估計量對半參數迴歸模型中非參數的部分進行估計。參數部分利用Newton-Raphson方法進行估計。
在本篇中，將核密度估計方法應用在不同分配之下的估計。並且比較不同帶寬、核函數的選擇對估計結果的影響。而在模擬研究中，假設未知累加基底風險函數服從韋伯分配，利用核密度估計方法估計出來的結果顯示，只要樣本數夠大，估計的表現較好。此外，核密度估計結果的好壞，與帶寬的選擇有密切的關係。

In survival analysis, the most commonly used models, the proportional hazard model and the proportional odds model, are special cases of linear transformation model. Because of its flexibility, our aim in this thesis is to explore the performance of kernel density estimation on unknown baseline cumulative hazard function under linear transformation model. In this thesis, we chose Nadaraya-Watson kernel estimator to estimate the nonparametric part of linear transformation model. Then we used Newton-Raphson method in the estimation of parametric part, and obtained the estimate of parameter which we are interested in.
We presented the application of kernel density estimation on different functions with different kernel functions and bandwidths. In simulation studies, we assume the baseline cumulative hazard function followed a Weibull distribution, and found that the result of kernel density estimation under different censored rate performed well when the sample size is large. We also found that the choice of bandwidth plays an important role in kernel estimation.

Directory
1. Introduction 1
1.1 Functions 2
1.2 Censored Data 3
1.3 Linear Transformation Model 4
1.4 Literature Review 5
1.5 Outline 6
2. The Application of Kernel Function 7
2.1 Standard Normal Distribution 13
2.2 Exponential Distribution 16
2.3 Weibull Distribution 19
3. Model and Parameter Estimation 22
3.1 Unknown Baseline Hazard Function 24
3.2 Newton-Raphson Method 26
4. Simulation Study 27
5. Conclusions 39
Reference 41
Appendix 43

 
List of Figures
Fig. 2.1 Histograms with different endpoints of bins 7
Fig. 2.2 Histogram with blocks centered over data points 8
Fig. 2.3 Estimated function with Gaussian kernel function 9
Fig. 2.4 Four kernel functions: Uniform, Gaussian, Triangle, Epanechnikov 10
Fig. 2.5 Four estimated functions with Gaussian kernel density estimation under the standard normal distribution. The blue line (dotted line) represents the bandwidth of 0.5 and the red line (dash line) represents the bandwidth of 0.05. 14
Fig. 2.6 Four estimated functions with Uniform kernel density estimation under the standard normal distribution. The blue line (dotted line) represents the bandwidth of 0.5 and the red line (dash line) represents the bandwidth of 0.05. 15
Fig. 2.7 Four estimated functions with Gaussian kernel density estimation under the exponential distribution. The blue line (dotted line) represents the bandwidth of 0.5 and the red line (dash line) represents the bandwidth of 0.05. 17
Fig. 2.8 Four estimated functions with Uniform kernel density estimation under the exponential distribution. The blue line (dotted line) represents the bandwidth of 0.5 and the red line (dash line) represents the bandwidth of 0.05. 18
Fig. 2.9 Four estimated functions with Gaussian kernel density estimation under the Weibull distribution. The blue line (dotted line) represents the bandwidth of 0.5 and the red line (dash line) represents the bandwidth of 0.05. 20
Fig. 2.10 Four estimated functions with Uniform kernel density estimation under the Weibull distribution. The blue line (dotted line) represents the bandwidth of 0.5 and the red line (dash line) represents the bandwidth of 0.05. 21
Fig. 4.1 Histograms of time t when the shape parameter (γ) of Weibull distribution is equal to 0.5, 1, 2 and 5. Sample size is 500. 29
List of Tables
Table 4.1 Kernel simulation 1 (n=300,ρ=1,α=2,γ=5) 30
Table 4.2 Kernel simulation 2 (n=300,ρ=1,α=2,γ=2) 30
Table 4.3 Kernel simulation 3 (n=300,ρ=1,α=2,γ=1) 31
Table 4.4 Kernel simulation 4 (n=300,ρ=1,α=2,γ=0.5) 31
Table 4.5 Kernel simulation 5 (n=300,ρ=1,α=1,γ=5) 32
Table 4.6 Kernel simulation 6 (n=300,ρ=1,α=1,γ=2) 32
Table 4.7 Kernel simulation 7 (n=300,ρ=1,α=1,γ=1) 33
Table 4.8 Kernel simulation 8 (n=300,ρ=1,α=1,γ=0.5) 33
Table 4.9 Kernel simulation 9 (n=500,ρ=1,α=2,γ=5) 34
Table 4.10 Kernel simulation 10 (n=500,ρ=1,α=2,γ=2) 35
Table 4.11 Kernel simulation 11 (n=500,ρ=1,α=2,γ=1) 35
Table 4.12 Kernel simulation 12 (n=500,ρ=1,α=2,γ=0.5) 36
Table 4.13 Kernel simulation 13 (n=500,ρ=1,α=1,γ=5) 36
Table 4.14 Kernel simulation 14 (n=500,ρ=1,α=1,γ=2) 37
Table 4.15 Kernel simulation 15 (n=500,ρ=1,α=1,γ=1) 37
Table 4.16 Kernel simulation 16 (n=500,ρ=1,α=1,γ=0.5) 38
Table A.1 Kernel simulation 17 (n=300,ρ=0,α=2,γ=5) 47
Table A.2 Kernel simulation 18 (n=300,ρ=0,α=2,γ=2) 48
Table A.3 Kernel simulation 19 (n=300,ρ=0,α=2,γ=1) 48
Table A.4 Kernel simulation 20 (n=300,ρ=0,α=2,γ=0.5) 49
Table A.5 Kernel simulation 21 (n=300,ρ=0,α=1,γ=5) 49
Table A.6 Kernel simulation 22 (n=300,ρ=0,α=1,γ=2) 50
Table A.7 Kernel simulation 23 (n=300,ρ=0,α=1,γ=1) 50
Table A.8 Kernel simulation 24 (n=300,ρ=0,α=1,γ=0.5) 51
Table A.9 Kernel simulation 25 (n=500,ρ=0,α=2,γ=5) 51
Table A.10 Kernel simulation 26 (n=500,ρ=0,α=2,γ=2) 52
Table A.11 Kernel simulation 27 (n=500,ρ=0,α=2,γ=1) 52
Table A.12 Kernel simulation 28 (n=500,ρ=0,α=2,γ=0.5) 53
Table A.13 Kernel simulation 29 (n=500,ρ=0,α=1,γ=5) 53
Table A.14 Kernel simulation 30 (n=500,ρ=0,α=1,γ=2) 54
Table A.15 Kernel simulation 31 (n=500,ρ=0,α=1,γ=1) 54
Table A.16 Kernel simulation 32 (n=500,ρ=0,α=1,γ=0.5) 55
Table A.17 Kernel simulation 33 (n=300,ρ=0.5,α=2,γ=5) 56
Table A.18 Kernel simulation 34 (n=300,ρ=0.5,α=2,γ=2) 57
Table A.19 Kernel simulation 35 (n=300,ρ=0.5,α=2,γ=1) 57
Table A.20 Kernel simulation 36 (n=300,ρ=0.5,α=2,γ=0.5) 58
Table A.21 Kernel simulation 37 (n=300,ρ=0.5,α=1,γ=5) 58
Table A.22 Kernel simulation 38 (n=300,ρ=0.5,α=1,γ=2) 59
Table A.23 Kernel simulation 39 (n=300,ρ=0.5,α=1,γ=1) 59
Table A.24 Kernel simulation 40 (n=300,ρ=0.5,α=1,γ=0.5) 60
Table A.25 Kernel simulation 41 (n=500,ρ=0.5,α=2,γ=5) 60
Table A.26 Kernel simulation 42 (n=500,ρ=0.5,α=2,γ=2) 61
Table A.27 Kernel simulation 43 (n=500,ρ=0.5,α=2,γ=1) 61
Table A.28 Kernel simulation 44 (n=500,ρ=0.5,α=2,γ=0.5) 62
Table A.29 Kernel simulation 45 (n=500,ρ=0.5,α=1,γ=5) 62
Table A.30 Kernel simulation 46 (n=500,ρ=0.5,α=1,γ=2) 63
Table A.31 Kernel simulation 47 (n=500,ρ=0.5,α=1,γ=1) 63
Table A.32 Kernel simulation 48 (n=500,ρ=0.5,α=1,γ=0.5) 64

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