The generalized sigmoidal quantile function

In this note we introduce a new smooth nonparametric quantile function estimator based on a newly defined generalized expectile function and termed the sigmoidal quantile function estimator. We also introduce a hybrid quantile function estimator, which combines the optimal properties of the classic kernel quantile function estimator with our new generalized sigmoidal quantile function estimator. The generalized sigmoidal quantile function can estimate quantiles beyond the range of the data, which is important for certain applications given smaller sample sizes. This property of extrapolation is illustrated in order to improve standard bootstrap smoothing resampling methods.