The geometric process is widely applied as a stochastic monotone model in many practical applications since its introduction. However, it sometimes does not satisfy some requirements in the real-world applications due to model limitations. For this reason, it is proposed a new stochastic model which is called doubly geometric process. In the applications of the doubly geometric process, the estimation problem associated with the process arises naturally. In this study, the statistical inference problem for the doubly geometric process is considered by assuming that the distribution of the first interarrival time has an exponential distribution. The maximum likelihood method is used to estimate the model parameters of the doubly geometric process and the parameter of distribution. The joint distribution of the maximum likelihood estimators is obtained. A simulation study is presented to evaluate the small sample performance of the estimators with different parameter values. Finally, three real-world-data sets are used to illustrate the applicability of the methods.