Best exact nonparametric confidence intervals for quantiles

Abstract

Well-known nonparametric confidence intervals for quantiles are of the form (X _{i : n} , X _{j : n} ) with suitably chosen order statistics X _{i : n} and X _{j : n} , but typically their coverage levels differ from those prescribed. It appears that the coverage level of the confidence interval of the form (X _{i : n} , X _{j : n} ) with random indices I and J can be rendered equal, exactly to any predetermined level γ ∈ (0, 1). Best in the sense of minimum E(J − I), i.e., ‘the shortest’, two-sided confidence intervals are constructed. If no two-sided confidence interval exists for a given γ, the most accurate one-sided confidence intervals are constructed.

Keywords:

• Confidence intervals
• Nonparametrics
• Quantiles