Estimation of Non-Unique Quantiles

Abstract

This paper is concerned with consistent estimates of a quantile of a distribution function when the quantile is not unique. To be more precise, since the quantile is assumed not to be unique, we are concerned with obtaining a consistent estimate of the smallest $p$th quantile for a fixed $p(0 < p < 1)$, and from this procedure we can estimate the largest $p$th quantile. In Section 2 we consider the oscillating character and limit distribution of the sample $p$th quantile. Also included in this section is a precise statement of the problem to be solved. In Section three the problem of medians only is considered. Here we treat the sample median of the set of averages of all $\binom{n + 1}{2}$ pairs of observations $X_1, \cdots, X_n$, which is briefly mentioned in [1]. We give a proof that this sample median converges almost surely to the center median of the original population, provided that the original distribution function is symmetric about a median. If this symmetry condition is relaxed, it is shown that this sample median of averages of pairs need not converge; and even if it did converge, it might converge to a number which is not a median of the parent distribution. In Section 4, strongly consistent estimates of the smallest $p$th quantile are obtained (for fixed $p, 0 < p < 1$) which do not depend on the functional form of the parent distribution function, and a characterization of weakly consistent estimates is given.