臺灣博碩士論文加值系統

Interval censoring refers to a situation in which, T, the time to occurrence of an event of
interest is only known to lie in an interval [L,R]. In some cases, the variable T also suers
left-truncation. Based on an integral equation, we propose a self-consistent estimator (SCE)
of survival function of T. It is shown that the NPMLE is a solution of the integral equation.
Under some conditions, we show the consistency of the SCE.

Content
1. Introduction......................................................................................1
Example 1: AIDS Cohort Studies.......................................................................1
2. The Nonparametric Estimators .....................................................................2
2.1 The NPMLE........................................................................................2
2.2 SCE..............................................................................................4
2.3.1 Theorem 1......................................................................................5
2.3.2 Theorem 2......................................................................................6
3. Simulation Results ...............................................................................8
3.1 Table 1..........................................................................................9
3.2 Table 2..........................................................................................9
3.3 Table 3.........................................................................................10
4. Discussions......................................................................................11
5.References........................................................................................11

Alioum A. and Commenges D. (1996). A proportional hazards model for arbitrarily censored
and truncated data. Biometrics, 52, 512-524.
Frydman, H. (1994). A note on nonparametric estimation of the distribution function from
interval-censored and truncated data. Journal of the Royal Statistical Society, Series B, 56,
71-74.
Gentleman, R. and Geyer C. J. (1994). Maximum likelihood for interval censored data:
consistency and computation. Biometrika, 81, 618-623.
Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maxi-
mum Likelihood Estimation. Basel: Birkhauser.
Gu, M. G. and Zhang, C. H. (1993), Asymptotic properties of self-consistent estimators
based on doubly censored data. The Annals of Statistics 21, 611-624.
Hudgens, M. G. (2005). On nonparametric maximum likelihood estimation with interval
censoring and truncation. Journal of the Royal Statistical Society, Series B, 67, part 4,
573-587.
Kalb
eish, J. D. and Lawless, J. F. (1989). Inferences based of retrospective ascertainment:
An analysis of the data on transfusion related AIDS. Journal of the American Statistical
Association, 84, 360-372.
Li, L., Watkins, T., Yu, Q. Q. (1997). An EM algorithm for smoothing the self-consistent
estimator of survival functions with interval-censored data. Scand. J. Statist., 24, 531-542.
Pan, W., Chappell, R. and Kosorok, M. R. (1998). On consistency of the monotone MLE of
13
survival for left truncated and interval-censored data. Statistics & Probability Letters. 38,
49-57.
Pan, W. and Chappell, R (1998). Computation of the NPMLE of distribution functions for
interval censored and truncated data with applications to the Cox model. Computational
Statistics and Data Analysis, 28, 33-50.
Pan, W. and Chappell, R (1999). A note on inconsistency of NPMLE of the distribution
function from left truncated and case I interval censored data. Lifetime Data Analysis, 5,
281-291.
Peto, R. (1973). Experimental survival curves for interval-censored data. Appl. Statist., 22,
86-91.
Shen, P.-S. (2005). Estimation of the truncation probability with the left-truncated and
right-censored data. Nonparametric Statistics, 17, No. 8, 957-969.
Shick, A and Yu, Q. 2000. Consistency of the GMLE with mixed case interval-censored
data. Scandinavian Journal of Statistics, 27, 45-55.
Song, S. (2004). Estimation with univariate \mixed case" interval censored data. Statist.
Sin., 14, 269-282.
Stvring, H. and Wang, M.-C. (2007). A new approach of nonparametric estimation of
incidence and lifetime risk based on birth rates and incident events. BMC Medical Research,
7:53, 1-11.
Thomas, D. R. and Grunkemeier, G. L. (1975). Condence interval estimation of survival
probabilities for censored data. Journal of the American Statistical Association, 70, 865-871.
Turnbull, B. W. (1974). Nonparametric estimation of a survivorship function with doubly
censored data. Journal of the American Statistical Association, 69, 169-173.
Turnbull, B. W. (1976). The empirical distribution function with arbitrarily grouped censored
and truncated data. Journal of the Royal Statistical Society, Series B, 38, 290-295.
van der Vaart, A. and Wellner, J. A. (2000). Preservation theorems for Glivenko-Cantelli
and uniform Glivenko-Cantelli classes. In High Dimensional Probability II, pp. 115-133.
Boston: Birkhauser.
Wang, M.-C. 1991, Nonparametric estimation from cross-sectional survival data. J. Amer.
Statist. Ass., 86, 130-143.
Woodroofe, M., 1985, Estimating a distribution function with truncated data. Ann. Statist.,
14
13, 163-167.
Yu, Q., Li, L. and Wong, G.Y.C., (1998a). Asymptotic variance of the GMLE of a survival
function with interval-censored data. Sankhya, 60, 184-187.
Yu, Q., Shick, A., Li, L. and Wong, G.Y.C., (1998b). Asymptotic properties of the GMLE
with case 2 interval-censored data. Statistics & probability letters, 37, 223-228.
Yu, Q. Q., Li, L., and Wong, G. Y. C. (2000), On consistency of the self-consistent estimator
of survival function with interval censored data. Scan. J. of Statist., 27, 35-44.
Yu, Q. Q., Wong, G. Y. C., and Li, L. (2001), Asymptotic properties of self-consistent
estimators with mixed interval-censored data. Ann. Inst. Stat. Math., 53, 469-486.