A Pointwise Estimator for the k-Fold Convolution of a Distribution Function

ABSTRACT

This article is concerned with some parametric and nonparametric estimators for the k-fold convolution of a distribution function. An alternative estimator is proposed and its unbiasedness, asymptotic unbiasedness, and consistency properties are investigated. The asymptotic normality of this estimator is established. Some applications of the estimator are given in renewal processes. Finally, the computational procedures are described and the relative performance of these estimators for small sample sizes is investigated by a simulation study.

Keywords:

• Convolution
• Empirical df
• Renewal processes
• V-statistics

Mathematics Subject Classification:

• Primary 62G05
• 62G20
• Secondary 60K05

Acknowledgments

The author is grateful to two anonymous referees and Professor N. Balakrishnan for their invaluable comments and suggestions for the improvement of this article.