Boundary-adaptive kernel density estimation: the case of (near) uniform density

We consider nonparametric kernel estimation of density functions in the bounded-support setting having known support $ [a,b] $ [a,b] using a boundary-adaptive kernel function and data-driven bandwidth selection, where a and b are finite and known prior to estimation. We observe, theoretically and in finite sample settings, that when bounds are known a priori this kernel approach is capable of outperforming even correctly specified parametric models, in the case of the uniform distribution. We demonstrate that this result has implications for modelling a range of densities other than the uniform case. Furthermore, when bounds $ [a,b] $ [a,b] are unknown and the empirical support (i.e. $ [\min (x_i),\max (x_i)] $ [min(xi),max(xi)]) is used in their place, similar behaviour surfaces.