题名 |
The Size of the Risk Set under Random Truncation |
并列篇名 |
左截資料下危險集合之大小 |
作者 |
沈葆聖(Pao-Sheng Shen) |
关键词 |
危險集合 ; 左截資料 ; risk set ; left-truncated data ; NPMLE |
期刊名称 |
統計與資訊評論 |
卷期/出版年月 |
15卷(2013 / 12 / 01) |
页次 |
47 - 55 |
内容语文 |
英文 |
中文摘要 |
Let U* and V* be two independent positive random variables with continuous distribution functions F and G. Let (af, ag) and (bf; bg) denote the lower and upper boundaries of U* and V* respectively. Under left truncation, both U* and V* are observable only when U*≥V*. Let (U1, V1),...,(Un, Vn) denote the truncated sample, and U(1)<U(2)<...<U(n) denote the distinct ordered statistics of the sample Ui's. Let (The equation is abbreviated.) denote the size of risk set. In applying the nonparametric maximum likelihood estimate (NPMLE) of distribution function F, a practical difficulty arises when Rn(U(i))=1 for some i≤n-1. Woodroofe (1985, Corollary 5) showed that when (F, G)єK, the probability P(Rn(U(i))=1 for some i≤n-1 converges to 0 as n→∞. In this note, we derive the exact probability of P(Rn(U(1))=k) for k=1,...,n and give an alternative proof of lim(subscript n→∞) P(Rn(U(1))=k)=0 for 1≤k<∞. Simulation results indicate that the probability P(Rn(U(1)=1) can be significant when af-ag is not sufficiently large. |
英文摘要 |
Let U* and V* be two independent positive random variables with continuous distribution functions F and G. Let (af, ag) and (bf; bg) denote the lower and upper boundaries of U* and V* respectively. Under left truncation, both U* and V* are observable only when U*≥V*. Let (U1, V1),...,(Un, Vn) denote the truncated sample, and U(1)<U(2)<...<U(n) denote the distinct ordered statistics of the sample Ui's. Let (The equation is abbreviated.) denote the size of risk set. In applying the nonparametric maximum likelihood estimate (NPMLE) of distribution function F, a practical difficulty arises when Rn(U(i))=1 for some i≤n-1. Woodroofe (1985, Corollary 5) showed that when (F, G)єK, the probability P(Rn(U(i))=1 for some i≤n-1 converges to 0 as n→∞. In this note, we derive the exact probability of P(Rn(U(1))=k) for k=1,...,n and give an alternative proof of lim(subscript n→∞) P(Rn(U(1))=k)=0 for 1≤k<∞. Simulation results indicate that the probability P(Rn(U(1)=1) can be significant when af-ag is not sufficiently large. |
主题分类 |
基礎與應用科學 >
資訊科學 基礎與應用科學 > 統計 |