Probabilities of Large Deviations of Sample Quantiles

Abstract

Let Y _n denote a suitably defined p-th (0 < p < 1) quantile computed from nk observations of which an equal number n of observations belong to each of a fui.ite number k of populations. Then, if _p is a correspondingly defined p-th population quantile, a is a positive number and l ⩾ 1 is any integer, it is shown that there exist $\rho n\hspace{0.17em}\left({\rho }_n\to \rho \hspace{0.17em}\text{as}\hspace{0.17em}n\to \infty ,0<\rho <1\right)\hspace{0.17em}\text{and}\hspace{0.17em}{m}_n\left(0<m_n<\infty \right)$ such that $P\left(Y_n-{\xi }_p⩽a\right)$ can be expressed as the product of $p_n^n{m}_n/{\left(2\pi n\right)}^{\frac{1}{2}}$ and a polynomial of degree l in n^-1 with a remainder term which is of order smaller than n^-l.