A simple hermitian estimator of the quantile function

The most commonly used quantile estimator, Q̂_L(u) = (1 - ∈)X_[n′u]:n + ∈X_[n′u]+1:n at (1.1) based upon the linear interpolation between two order statistics, lacks smoothness, and because of its nondifferentiability is not always usable to estimate the density-quantile function. In this note a monotone cubic version of the Hermite interpolation formula is used to construct and study a smooth quadratic estimator of the entire quantile function Q̂_H(u) = w₁X_{[n′u-1/2]:n} + w₂X_{[n′u-1/2]+1:n} + w₃X_{[n′u-1/2]+2:n}, (0.1) presented at (2.8), and the associated estimator [Q̂_H'(u)]^-1 for the density quantile function. These are simple enough to replace Q̂_L(u) and [Q̂_L'(u)]^-1 even where the latter does not exist. The asymptotic expansions up to O(n^-3) for the expectation and variance for Q̂_H(u) are provided. As special cases the Hermitian medians and Hermitian quartiles are examined. It is shown that an apparent anomaly of the variance of Q̂_L(1/2) considered by Hodges [1] and Hodges and Lehmann [2] is alleviated for the Hermitian median Q̂_H(1/2).