A New Proof of the Bahadur Representation of Quantiles and an Application

Let \Xᵢ\ be a sequence of independent random variables with the same distribution function F (x) = P r\Xᵢ x\. Let F (Mₚ) = p, 0 x. Bahadur (1966) has proved equation*1Y₏, ₍ = Mₚ + Gₙ (Mₚ) - (1 - p) /F' (Mₚ) + Rₙequation* where the remainder term Rₙ becomes negligible as n. More precisely, he has shown Rₙ = O (n^-3{4} n) a. s. as n. The best result of this type is due to Kiefer (1967) who has calculated the exact order of Rₙ. Sen (1968) has extended Bahadur's result to random variables which are neither independent nor identically distributed. We shall give a new and much simpler proof of a weaker version of Bahadur's result which suffices for many statistical applications. Our proof involves fewer assumptions than Bahadur's. For arbitrary pₙ let M₏䂸 be defined as Mₚ + (pₙ - p) /F' (Mₚ). Consider equation*2Y₏䂸, ₍ = M₏䂸 + Gₙ (Mₚ) - (1 - p) /F' (Mₚ) + Rₙequation* where Y₏䂸, ₍ is a sample pₙ-quantile. In Section 2 we have proved the following result about Rₙ. THEOREM 1. Suppose F' (Mₚ) exists and is strictly positive and pₙ - p = O (1/n^1{2}). Then Rₙ as defined in (2) (and, a fortiori, Rₙ as defined in (1) satisfies equation*3n^1{2} Rₙ 0 in probability. equation* (After writing this paper the author discovered that the result for pₙ = p is stated without proof in Chernoff et al (1967). ) It is easy to extend this result as in Sen (1968). An outline is sketched in one of the remarks. Once again it is possible to achieve some economy in assumptions. The representation (1) is not new. Its use in deriving the asymptotic moments of Y₏, ₍ goes back to Karl Pearson. See, for example, (1) in Hojo (1931). But the formulation therein is very imprecise and lacks a rigorous justification. We next consider an application of Theorem 1. Let Xₙ = (ⁿ₁ Xᵢ) /n and Pₙ = proportion of Xᵢ's above Xₙ. David (1962) proved the asymptotic normality of Pₙ when F is a normal distribution function. Using the same elegant trick, Mustafi (1968) has proved a similar result for bivariate normal distributions. We shall extend these results considerably by providing alternative proofs based on Theorem 1, which dispense with the normality assumption on F. Moreover, in our proof we may consider--though we shall not do so for purposes of simplicity--instead of the sample mean Xₙ an U-statistic to which the central limit theorem of Hoeffding (1948) applies.