A Note on the Accuracy of Normal Approximation of Random Quantities

摘要

For a sequence of independent, identically distributed random variable (iid rv's) { X n } and a sequence of integer-valued random variables { N n } , define the random quantiles as ξ ̂ N n , p = X ( [ N n p ] + 1 ) , where [ x ] denote the largest integer less than or equal to x , and X ( i ) the i th order statistic in a sample X 1 , X 2 , . . . , X n and 0 ≤ p ≤ 1 . In this note, the limiting distribution and its exact order approximation are obtained for ξ ̂ N n , p . The limiting distribution result we obtain extends the work of several including Wretman [ 1 ] . The exact order of normal approximation generalizes the fixed sample size results of Reiss [ 2 ] . AMS 2000 subject classification: 60F12; 60F05; 62G30.