Exponential convergence for the k-th order statistics

Abstract

Let $X_1, X_2, \ldots, X_n$ be the samples of an arbitrary
population having a uniform distribution on the interval $[0,1]$ and $X_{1,n}\le X_{2,n}\le \cdots\le X_{n,n}$ denote the order statistics. The moderate and large deviations for the $k$-th order statistics $X_{k,n}$ are established. In addition, we are considering the moderate and large deviations of $F(X_{k,n})$, where $F$ is the cumulative distribution function of $X_1,X_2,\ldots, X_n$.