Consistency of the beta kernel density function estimator

By Taoufik Bouezmarni, J. Rolin. 2003

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Probit Transformation for Kernel Density Estimation on the Unit Interval (2013, methodology): “The asymptotic bias and variance of the Beta kernel estimator were given in Chen (1999), and further asymptotic properties were studied in Bouezmarni and Rolin (2003) and Zhang and Karunamuni (2010).”
Smooth estimation of a distribution and density function on a hypercube using Bernstein polynomials for dependent random vectors (2006, background): “Chen (1999) introduced a beta kernel density estimator for random variables with compact support on ½0; 1 ; and recently Bouezmarni and Rolin (2003) have investigated its consistency properties.”
Bank loan recovery rates: Measuring and nonparametric density estimation (2010, background): “Moreover, Bouezmarni and Rolin (2003) prove that, even if the unknown density function f is infinite at a boundary, such estimator remains”
Miscellanea kernel-type density estimation on the unit interval (2007, background): “(For more on asymptotic theory for f C1 , see Bouezmarni & Rolin, 2003.)”
Hausdorff moment problem: Reconstruction of probability density functions (2008, background): “(α − [αx] − m)! f (t) dt = ∫ 1 0 β(t, [αx] + 1, α − [αx] + 1) f (t)dt. (12) Now one can derive fα,ν −→u f from the last line in (12) (see, Bouezmarni and Rolin (2003), or Feller (1971), Ch. VII)), since {β(·, [αx]+1, α−[αx]+1), α ∈ N} forms a δ-sequence at x , as α → ∞, its mean ηα = ([αx]+1)/(α+2)…”