| Abstract: | 雙截取資料在天文學中扮演了一個很重要的角色，如同在存活分析中一樣。分別以(U_i*,V_I*,Y_i*,Z_i*）表示左截取,右截取,存活時間及共變異數，其中U_i*及V_i*會受到Z_i*的影響，而存活時間需落在左右截取中才能觀察得到。假設在存活時間與左右截取時間為獨立的情況下，在無母參數最大概似估計(NPMLE)之下，存活時間會有一個失效函數，這是Efron和Petrosian在1999所提出的，我們的論文主要是使用另一種方法推導出失效函數。
Doubly truncated data play an important role in the statisitcal analysis of astronomical observations as well as in survival analysis. Let (U_1* , V_1* , Y_1* ,Z_1* ), (U_2* , V_2* , Y_2* ,Z_2* ), . . . be i.i.d. continuous random vectors, where U_i* , V_i* , Y_i* and Z_i* represent left-truncation time, right-truncation time, lifetime and a covariate, respectively. Both U_i* and V_i* depend on Z_i* . Under double truncation models, a quadruple of independent variable (U_i* , V_i* , Y_i* ,Z_i* ) is observable only when U_i* <= Y_i* <= V_i* . Under the assumption that Y_i* is independent of (U_i* , V_i* ), the nonparametric maximum likelihood estimate (NPMLE) of hazard function of Y_i* (denoted by ˆh) was developed by Efron and Petrosian (1999). In this note, we present an alternative approach to derive ˆh. |
