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In this study, the mean value and variance functions of type II counter process constituted as a delayed renewal process are obtained analytically. In the applications of the delayed renewal process, generally, the functional forms of the distributions are known but some parameters of the distributions are unknown. In those cases, the mean value and the variance functions must be estimated from the data. In this study, after obtaining the mean value and variance functions of type II counter process analytically, depending on the analytical expressions, we deal with the problem of estimating these functions and some estimators are proposed. Performance of the estimators is evaluated for small sample sizes by a simulation study according to bias and mean square error criteria. [ABSTRACT FROM AUTHOR]