Asymptotic distributions of non-central studentized statistics

Let X ₁, ..., X _n be independent and identically distributed random variables and W _n = W _n (X ₁, ..., X _n ) be an estimator of parameter ϑ. Denote T _n = (W _n − ϑ ₀)/s _n , where s _n ² is a variance estimator of W _n . In this paper a general result on the limiting distributions of the non-central studentized statistic T _n is given. Especially, when s _n ² is the jacknife estimate of variance, it is shown that the limit could be normal, a weighted χ ² distribution, a stable distribution, or a mixture of normal and stable distribution. Applications to the power of the studentized U- and L- tests are also discussed. Shao’s work was supported in part by Hong Kong UST (Grant No. DAG05/06.SC) and Hong Kong RGC CERG (Grant No. 602206), and Zhang’s work partially supported by National Natural Science Foundation (Grant No. 10801118), and the PhD Programs Foundation of the Ministry of Education of China (Grant No. 200803351094)