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.

Abstract 2
1 INTRODUCTION 2
2 AN ALTERNATIVE APPROACH 4
3 SIMULATION RESULTS 10
BIBLIOGRAPHY 12

Bilker, W. B. and Wang, M.-C. A semiparametric extension of the Mann-Whitney
test for randomly truncated data. Biometrics, 1996, 52, 10-20.
Efron, B. and Petrosian, V. Noparametric methods for doubly truncated data. J.
Amer. Statist. Assoc., 1999, 94, 824-834.
Kaplan, E. L. and Meier, P. Nonparametric estimation from incomplete observations,
J. Amer. Statist. Assoc., 1958, 53, 457-481.
Lynden-Bell, D. A method of allowing for known observational selection in small
samples applied to 3CR quasars. 1971, Mon. Not. R. Astr. Soc. 155, 95-118.
McLaren, C.,Wagstaff, M., Brittegram, G., and Jacobs, A. Detection of two-component
mixtures of lognormal distributions in grouped doubly truncated data analysis of red
blood cell volume distributions. Biometrics, 1991, 47, 607-622.
Statten, G. A. and Datta S. The Kaplan-Meier estimator as an inverse-probabilityof-
censoring weighted average. Amer. Statist. Ass., 2001, 55, 207-210.
Turnbull, B. W. The empirical distribution with arbitrarily grouped, censored and
truncated data. J. R. Statist. Soc. B., 1976, 38, 29
Shen, P-S. The product-limit estimate as an inverse-probability-weighted average.
Communi. in Statist., Part A- Theory and Methods, 2003, 32, 1119-1133.
Woodroofe, M. Estimating a distribution function with truncated data. 1985, The
Annals of Statistics, 13, 163-167.
Wang, M.-C. Product-limit estimates: a generalized maximum likelihood study. Communi.
in Statist., Part A- Theory and Methods, 1987, 6, 3117-3132.