Kernel Quantile Estimators

SUMMARY The estimation of population quantiles is of great interest when one is not prepared to assume a parametric form for the u.nderlying distribution. In addition, quantiles often arise as the natural thing to estimate when the underlying distribution is skewed. The sample quantile is a popular nonparametric estimator of the corresponding population quantile. Being a function of at most two order statistics, sample quantiles experience a substantial loss of efficiency for distributions such as the normal. An obvious way to improve efficiency is to form a weighted average of several order statistics, using an appropriate weight function. Such estimators are called L-estimators. The problem then becomes one of choosing the weight function. One class of L-estimators, which uses a density function (called a kernel) as its weight function, are called kernel quantile estimators. The effective performance of such estimators depends critically on the selection of a smoothing parameter. An important part of this paper is a theoretical analysis of this selection. In particular, we obtain an expression for the value of the smoothing parameter which minimizes asymptotic mean square error. Another key feature of this paper is that this expression is then used to develop a practical data-based method for smoothing parameter selection. Other L-estimators of quantiles have been proposed by Harrell and Davis estimator is just a bootstrap estimator (Section 1). An important aspect of this paper is that we show that asymptotically all of these are kernel estimators with a Gaussian kernel and we identify the bandwidths. It is seen that the choices of smoothing parameter inherent in both the Harrell and Davis estimator and the Brewer estimator are asymptotically suboptimal. Our theory also suggests a method for choosing a previously not understood tuning parameter in the Kaigh-Lachenbruch estimator. The final point is an investigation of how much reliance should be placed on the theoretical results, through a simulation study. We compare one of the kernel estimators, using data-based bandwidths, with the Harrell-Davis and Kaigh-Lachenbruch estimators. Over a variety of distributions little consistent difference is found between these estimators. An important conclusion, also made during the theoretical analysis, is that all of these estimators usually provide only modest improvement over the sample quantile. Our results indicate that even if one knew the best estimator for each situation one can expect an average improvement in efficiency of only 15%. Given the well-known distribution-free inference procedures (e.g., easily constructed …