臺灣博碩士論文加值系統

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

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