Some asymptotic properties of smooth quantile ratio estimation.

摘要

Smooth quantile ratio estimation (SQUARE) is a robust method for estimating the difference between two means. Although the two samples are assumed to be independent, the distributions are taken to be linked. Specifically, the log ratio of the two quantile functions is assumed to be a smooth function lying in a specified function space, defined by an orthonormal set of basis functions.; Smooth quantile ratio estimation is close to the theory of L-statistics, and may be thought of as a specific semi-parametric extension of that theory, where the score function J(p) is stochastic instead of deterministic, depending on the empirical quantile function.; The SQUARE estimator is shown to be consistent and asymptotically normal under conditions very similar to those under which G. Shorack and J. Wellner proved the consistency and asymptotic normality of L-statistics.; The basic results are extended to include certain practical steps one might take when implementing the SQUARE algorithm. These variations on the algorithm lead to estimates with the same asymptotic distribution as the original version. A further extension allows for the estimation of the log quantile ratio function within a larger function space. In this version, elements of an infinite set of orthonormal functions are added slowly to an active basis as sample size increases.