Estimates for Order Statistics in Terms of Quantiles

Let X 1 ,..,X n be independent nonnegative random variables with cumulative distribution functions F 1 , F 2 ,.. , F n satisfying certain (rather mild) conditions. We show that the median of kth smallest order statistic of the vector (X 1 ,.. , X n ) is equivalent to the quantile of order (k − 1/2)/n with respect to the averaged distribution F=1n∑i=1nFi. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.