Estimates for Order Statistics in Terms of Quantiles

Resumo

Let X₁, . . .,X_n be independent nonnegative random variables with cumulative distribution functions F₁, F₂, . . . , F_n satisfying certain (rather mild) conditions. We show that the median of kth smallest order statistic of the vector (X₁, . . . , X_n) is equivalent to the quantile of order (k − 1/2)/n with respect to the averaged distribution \( F=\frac{1}{n}\sum \limits_{i=1}^n{F}_i \).