Some Normal Approximations for Renewal Function of Large Weibull Shape Parameter

Abstract

Weibull renewal function has attracted a lot of attention because the Weibull distribution describes in a relatively simple analytical manner a wide range of realistic behavior and its shape and scale parameters can be readily determined with graphical or statistical procedure. On the other hand, there are no closed form analytical solutions for the Weibull renewal function except for the special case of exponential distribution. Bounds, approximations, and tables are usually used. In this article, some approximations based on Normal approximation of Weibull distribution are studied. Such a procedure, which is different from that in the existing literature, is shown to be good for Weibull renewal function when the shape parameter is of moderate or large size. Series truncation expression and approximation bounds can be obtained as well. Numerical examples and comparisons are shown to illustrate the procedure.

Keywords:

• Renewal function
• Weibull distribution
• Normal approximation
• Shape parameter
• Series truncation approximation

Acknowledgments

We would like to thank Prof N. I. Robinson for several useful comments on an early draft of this article. This research is supported by the National University of Singapore under the research grant RP3992679.