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In this study, we deal with the estimation problem of the trend renewal function which is given as the mean value function of the trend renewal process by assuming that the distribution function of the time-transformed random variables is a gamma distribution with the shape parameterand the scale parameterFor this purpose, to begin with, we yield the calculation procedure for the trend renewal function by using the Riemann-Stieltjes method. Then, we estimate the model parameters of the trend renewal process. Based on these estimators, an estimator of the trend renewal function is proposed and consistency property of this estimator is investigated. A simulation study is performed to evaluate the performance of all the given estimators according to bias and mean square error criteria. Finally, the applicability of the estimation procedure is applied to two real data sets.

Estimation of the mean value function for gamma trend renewal process