A Note on the Accuracy of Normal Approximation of Random Quantities

Abstract

For a sequence of independent, identically distributed random variable (iid rv's) $\left\{X_n\right\}$ and a sequence of integer-valued random variables $\left\{N_n\right\}$ , define the random quantiles as ${\widehat{\xi }}_{N_n,p}=X_{([N_{n}p]+1)}$ , 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 $0\le p\le 1$ . In this note, the limiting distribution and its exact order approximation are obtained for ${\widehat{\xi }}_{N_n,p}$ . 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.