A Note on the Accuracy of Normal Approximation of Random Quantities. Issue 1 (May 2021) - Ahmad, Ibrahim A.; Mugdadi, A. R.
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 asξ ̂ N n, p = X ( [ N n p ] + 1 ), where[ x ] denote the largest integer less than or equal tox, andX ( i ) thei th order statistic in a sampleX 1, X 2, . . ., X n and0 ≤ p ≤ 1 . In this note, the limiting distribution and its exact order approximation are obtained forξ ̂ 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.