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研究生:陳俊男
研究生(外文):Chun-Nan Chen
論文名稱:以逆加權方法估計截取機率及設限分配
論文名稱(外文):An inverse-probability-weighted approach to estimation of truncation probability and censoring distribution
指導教授:沈葆聖沈葆聖引用關係陳臺芳
指導教授(外文):Pao-Sheng ShenTai-Fang Chen
學位類別:碩士
校院名稱:靜宜大學
系所名稱:應用數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2005/07/
畢業學年度:93
語文別:英文
論文頁數:22
中文關鍵詞:左截取右設限截取機率
外文關鍵詞:right censoringtruncation probability.Left truncation
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本文在探討在左截取右設限的資料中, 我們以F, Q 和G分別來表示Ui*,Ci*和Vi*的分配函數, 當Ui*< Vi*的情況下, 我們無法收集到任何資料; 但當Ui*≧ Vi*時我們可收集到(Xi *; δi*), 其中Xi *= min(Ui*; Ci*) 且δi* = I[Ui*≦Ci*]。我們將探討如何估計截取機率α = P(Ui*≧Vi* )和設限分配Q。在P(Ci*≧Vi* ) = 1的條件下,1991年Wang 提出用α
= ∫[1-Fn(s-)]dGn(s)來估計α, 這裡的Fn 和Gn是F和G的NPMLE。本論文中, 我們將用逆加權方法(IPW) 估得α
的另一種表示方法α^n。在α^n, Fn and Gn的好的性質下我們用IPW 來估Q (記為Q^e)。在中等樣本數下Q^e 和 α
在也會有好的性質。
Let (Ui* ,Ci*, Vi* ) be i.i.d. random vectors such that (Ci*, Vi*) is independent of Ui* and P(Ci*≧Vi* ) = 1. Let F, Q and G denote the common distribution function of Ui* ,Ci* and Vi* , respectively. For left-truncated and right-censored data, one can observe nothing if Ui*&lt; Vi* and observe (Xi *; δi*), with Xi *= min(Ui*; Ci*) andδi* = I[Ui*≦Ci*], if Ui*≧ Vi*. Two questions of interest are how to estimate the truncation probabilityα = P(Ui* ≧Vi* ) and the censoring distribution Q. Under the constraint that P(Ci*≧Vi* ) = 1, Wang (1991) suggested estimating α byα
= ∫[1-Fn(s-)]dGn(s), where Fn and Gn are nonparametric maximum likelihood estimate (NPMLE) of the distributions F and G, respectively. In this note, using an inverse-probability-weighted (IPW) approach, we obtain an alternative representation α^n for α
. With this, good behaviors ofα^n, Fn and Gn induce nice properties in the IPW estimator of q (denoted by Q^e). Simulation study shows that both Q^e and α
work satisfactorily for moderate sample size.
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Inverse-probability-weighted Estimators . . . . . . . . . . . . . . . . 8
2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 The NPMLE of F, G and Q . . . . . . . . . . . . . . . . . . . . . . . 9
3 The Equivalence of n, ˜n and ˆn . . . . . . . . . . . . . . . . . .12
4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
[1] Gross, S. T. and Lai, T. L. Nonparametric estimation and regression analysis with left-truncated and right-censored data. J. Amer. Statist. Ass., 1996, 91,1166-1180.”
[2] Gross, S. T. and Lai, T. L. Boostrap methods for truncated data and censored data. Statist. Sinica, 1996, 6, 509-530.”
[3] He, S. and Yang, G. L. Estimation of the truncation probability in the random truncation model. Ann. Statist., 1998, 26, 1011-1027.
[4] Satten, G. A. and Datta S. The Kaplan-Meier estimator as an inverse-probabilityof-censoring weighted average. Amer. Statist., 2001, 55, 207-210.
[5] Shen, P.-S. The product-limit estimates as an inverse-probability-weighted average.Communi. in Statist., Part A- Theory and Methods, 2003, 32, 1119-1133.
[6] Tsai, W.-Y., Jewell, N. P. and Wang, M.-C. A note on the product-limit estimate under right censoring and left truncation. Biometrika, 1987, 74, 883-886.
[7] Wang, M.-C.; Jewell, N. P.; Tsai, W.-Y. Asymptotic properties of the productlimit estimate under random truncation. Ann. Statist., 1986, 14 1597-1605.
[8] Wang, M.-C. Product-limit estimates: a generalized maximum likelihood study. Communi. in Statist., Part A- Theory and Methods, 1987, 6, 3117-3132.
[9] Wang, M.-C. Nonparametric estimation from cross-sectional survival data. J.
Amer. Statist. Ass., 1991, 86, 130-143.
[10] Woodroofe, M. Estimating a distribution function with truncated data. Ann.
Statist., 1985, 13, 163-167.
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