The asymptotic distribution of order statistics
The asymptotic distribution of order statistics
复制标题
阶次统计量的渐近分布
DOI：
10.1002/nav.3800260307
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发表时间：
1979
期刊：
影响因子：
--
通讯作者：
L. Weiss
中科院分区：
文献类型：
--
作者：
L. Weiss
对于每个n，X1(N)，X2(N)，…，Xn(N)是IID，具有公共pdf Fn(X)。Y1(N)<；…<；Yn(N)是X1(N)，…的有序值、Xn(N)。Kn是正整数，其中LIM Kn=∞。在Kn和Fn(X)的某些条件下，以前的一篇文章证明了变量Y1(N)，…的一个特殊集合Kn+1的联合分布，Yn(N)对于所有渐近概率计算都可以假设为正态。在另一篇文章中，证明了如果Fn(X)随着n的增加以一定的速度逼近在(0，1)上一致的pdf，那么对于所有的渐近概率计算，不在特殊集合中的顺序统计量的条件分布可以被假定为一致的。本文证明了，即使Fn(X)不随n的增加而接近均匀分布，对于所有渐近概率计算，特殊集合中的顺序统计量之间包含的顺序统计量的分布也可以假定为均匀随机变量的二次函数的分布。给出了在统计推断中的应用。
For each n., X1(n), X2(n), …, Xn(n) are IID, with common pdf fn(x). y1(n) < … < Yn (n) are the ordered values of X1 (n), …, Xn(n). Kn is a positive integer, with lim Kn = ∞. Under certain conditions on Kn and fn (x), it was shown in an earlier paper that the joint distribution of a special set of Kn + 1 of the variables Y1 (n), …, Yn (n) can be assumed to be normal for all asymptotic probability calculations. In another paper, it was shown that if fn (x) approaches the pdf which is uniform over (0, 1) at a certain rate as n increases, then the conditional distribution of the order statistics not in the special set can be assumed to be uniform for all asymptotic probability calculations. The present paper shows that even if fn (x) does not approach the uniform distribution as n increases, the distribution of the order statistics contained between order statistics in the special set can be assumed to be the distribution of a quadratic function of uniform random variables, for all asymptotic probability calculations. Applications to statistical inference are given.