Exponential Convergence for the $k$-th Order Statistics

Suxia Yao, Yu Miao, Saralees Nadarajah

Let $X_1,X_2,\cdots,X_n$ be the samples of an arbitrary population having a uniform distribution on the interval $[0,1]$ and $X_{1,n}\leq X_{2,n}\leq\cdots\leq X_{n,n}$ denote the order statistics. The moderate deviations, large deviations and Cramér large deviations for the $k$-th order statistics $X_{k,n}$ are established. Furthermore, we also discuss the case that the samples come from the general distribution $F$.