Probability density function estimation by positive quartic C2-spline functions

Spline is a continuous function piecewise-defined by polynomials and is widely used for interpolation and smoothing of observed data. In 1994, Heβ and Schmidt proposed a positive quartic C
²
-spline interpolation for estimation of a non-negative and twice continuously differentiable function. In this paper, first we generalize the positive quartic C
²
-spline interpolation to the positive quartic C
²
-spline smoothing. Then we propose two estimation methods of a probability density function from its histogram by extending the ideas of the positive quartic C
²
-spline interpolation and smoothing. Finally numerical experiments show the effectiveness of the proposed methods.