For modeling extremal behaviors, the Generalized extreme value distribution that originated from the well established Extreme Value Theory has been widely used. As a special case of such Generalized extreme value distribution, the Gumbel family is suitable for modeling maximum values from light-tailed distributions. A common assumption used in the central models of extreme values is the independence of extremes in most previous studies. However,
short-term dependence among extremes might exist. In this thesis, we study a linear Gumbel distributed autoregressive model which was introduced by Toulemonde et al. (2010) to simulate dependent extremes that follow the Gumbel distribution. Our main goal is to investigate that if Gumbel distributed short-term maxima are weakly/moderately/strongly dependent, but this dependence is not recognized, what will happen to the resulting estimates of the Gumbel parameters. To reach this goal, simulations and a numerical example in
environmental science are presented to quantify the above issue.