AISM 52, 471-480

## Probability density function estimationusing 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, \infty) 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 non-negative 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.

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