When is the Student $t$-statistic asymptotically standard normal?
When is the Student $t$-statistic asymptotically standard normal?
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DOI：
10.1214/aop/1024404523
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发表时间：
1997-07
影响因子：
2.3
通讯作者：
E. Giné;F. Götze;D. Mason
中科院分区：
文献类型：
--
作者：
E. Giné;F. Götze;D. Mason
设X，Xi，i ∈ N是独立同分布的随机变量.证明了基于样本{Xi} ni =1的Student t-统计量渐近N（0，1）当且仅当X在正规律的吸引域中。还证明了对任意X，如果自归一化和Un：=<$ni = 1Xi/（<$ni = 1Xi 2）1/2，n ∈ N是随机有界的，则它们是一致次高斯的，即supnEexp（λ Un 2）0。
Let X, X i , i ∈ N, be independent, identically distributed random variables. It is shown that the Student t-statistic based upon the sample {X i } n i =1 is asymptotically N(0,1) if and only if X is in the domain of attraction of the normal law It is also shown that, for any X, if the self-normalized sums U n := Σn i=1 X i /(Σ n i=1 X i 2 ) 1/2 , n ∈ N, are stochastically bounded then they are uniformly subgaussian that is, sup n E exp(λU n 2 ) 0.