Bahadur–Kiefer Type Representations for Smoothed Conditional Quantile Estimators

Abstract

Bahadur and Kiefer derived almost sure (a.s.) representations for the (unconditional) sample quantile function in terms of the standard (unsmoothed) empirical distribution function. Their representations later became commonly known as the Bahadur–Kiefer (BK) representations. In this article, we establish BK type a.s. representations, and the resulting laws of iterated logarithm, for three distinct fully nonparametric smooth conditional quantile estimators—with optimal orders for the remainders—viz. for a smooth linear type, a Parzen-type smoothed (integrated) inverse and a smooth inverse type (kernel) conditional quantile estimator (c.q.e.) under some broad conditions on the underlying cdf’s and the kernels and bandwidth sequences employed. We also demonstrate that of these the linear type c.q.e. is, in fact, ‘second-order-equivalent’ to the Parzen-type smoothed (integrated) inverse c.q.e. Some remarks are included on the comparative merits of these smooth c.q.e.’s, and their BK representations relative to their smooth and unsmoothed counterparts studied earlier in literature and possible extensions of the present results. Our results are of the exact a.s. type and provide improvements over those achieved hitherto in literature. They are of considerable value for studying the asymptotics of quantile regression analytics.

AMS Subject Classification: Primary 62G05, 62G07; secondary: 60F15, 62G20, 62G30