| Abstract: | In statistical analysis the quantile function (qf) and the quantile density function (qdf) are equally important as the cdf and the pdf. Numerous smooth quantile function estimators have been proposed as alternatives to the sample quantile function. The research in this area has been somewhat esoteric in the sense that little work has been done in terms of providing guidelines for practice. This research is motivated by several practical issues involved in quantile function estimation. In the attempt to compare various qf estimators and to provide guidelines for their uses in practice, asymptotic behavior of translation and scale equivariant qf estimators and the derived qdf estimators are first investigated under a unified representation. It is shown that under moderate conditions these estimators share equally good asymptotic behavior. Some existing asymptotic results are strengthened and extended. Sufficient conditions for the consistency of the derived qdf estimators are obtained as well. The practical impact of the asymptotic results is the demonstration of the fact that smoothing the sample quantile function by any reasonable kernel always results in a qf estimator possessing nice asymptotic behavior. So asymptotic theory is not informative for obtaining guidelines for the selection of kernels in practice (finite samples). Finite-sample behavior of several translation and scale equivariant qf estimators is then formulated and compared according to several criteria. The finite sample results, in conjunction with the asymptotics, clearly suggest certain smoothing kernels to use in practice. A data-driven procedure to determine amount of smoothing is proposed. State-of-the-art demonstrations of the quantile-domain data analysis methodology are presented by analyzing several well-known data sets. |
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