The Generalized Extreme Value (GEV) distribution in the Extreme Value Theory (EVT) has been widely applied to analyze various extreme values. The Gumbel, one of the three families of GEV, is a light-tailed distribution family. Since the independence among extremes rarely holds in application, dependent models are more practical. This thesis focuses on a temporally dependent model, speci cally the Gumbel autoregressive model of order 1. We proposed a special method of moments estimation for the location and the scale parameters based on a link between Exponential distribution and Gumbel distribution. The existence and uniqueness of the estimates have been proven. Along with the least squares method for the autoregressive coe cient, we investigated the performance of those estimators and compared them with the combination of the Yule-Walker estimator and the ordinary method of moments estimators, which was used in an environmental study conducted by Toulemonde
et al.(2010). The average run length (ARL) of the model is studied through simulation. Two
examples of real data are used to illustrate our methodology.