PROBABILITY DENSITY FUNCTION ESTIMATION USING GAMMA KERNELS

SONG XI CHEN*

Department of Statistical Science, La Trobe University, Victoria 3083, Australia

(Received May 25, 1998; revised April 12, 1999)

Abstract. We consider estimating density functions which have support on [0, ∞) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are nonnegative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n^{−4/5}x^{−1/2}) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators.

Key words and phrases: Boundary bias, gamma kernels, local linear estimators, variable kernels.