Bias reduction by transformed flat-top Fourier series estimator of density on compact support

The problem of nonparametric estimation of a univariate density with rth continuous derivative on compact support is addressed ( $ r\geq 2 $ r≥2). If the density function has compact support and is non-zero at either boundary, regular kernel estimator will be completely biased at such boundary. Although several correction methods were proposed to improve the bias at the boundary to $ h^2 $ h2 in the last decades, this paper initiates a way to further improve bias to higher order ( $ h^r $ hr) for interior area of density function support, while remaining the order of bias $ h^2 $ h2 at boundary. We will first review flat-top kernel estimator and flat-top series estimator, then propose the Transformed Flat-top Series estimator. The theoretical analysis is supplemented with simulation results as well as real data applications.