Under the assumption of a uniform parent distribution, we find an exact expression for the mean-square error of an order-statistic approximation suggested by R. R. Bahadur. From this result we obtain an asymptotic formula for the mean-square error when the parent distribution is not necessarily uniform by transforming with the inverse of the cumulative distribution function of the parent.
"The Mean-Square Error of Bahadur's Order-Statistic Approximation." Ann. Statist. 1 (3) 446 - 453, May, 1973. https://doi.org/10.1214/aos/1176342410