Abstract
In this paper, an integral equation for the kth moment function of a geometric process is derived as a generalization of the lower-order moments of the process. We propose a general solution to solve this integral equation by using the numerical method, namely trapezoidal integration rule. The general solution is reduced to the numerical solution of the integral equations which will be given for the third and fourth moment functions to compute the skewness and kurtosis of a geometric process. To illustrate the numerical method, we assume gamma, Weibull and lognormal distributions for the first interarrival time of the geometric process.
Highlights
The geometric process (GP), which is a natural generalization of a renewal process (RP), is an important stochastic monotone model used in many areas of statistics and applied probability, especially for statistical analysis of series of events
We present an integral equation for the kth moment function as a generalization of the lower-order moment functions of the GP
A numerical method established for solving the integral equation given for the kth moment function is presented
Summary
The geometric process (GP), which is a natural generalization of a renewal process (RP), is an important stochastic monotone model used in many areas of statistics and applied probability, especially for statistical analysis of series of events. By considering the distribution functions of the interarrival times, one of the important di¤erences between RP and GP can be given as follows. Many researchers and authors made some studies on the ...rst and second moment functions of the GP by considering these lifetime distributions. A generalization of the integral equation for kth moment function of the GP is given. We adapt the Tang and Lam’s numerical method for the kth moment function of the GP with the help of the integral equation given for this function.
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