In extreme value theory, the generalized extreme value (GEV) distribution and the generalized Pareto (GP) distribution are the main statistical models widely used to characterize extreme values of univariate random variables having unknown probability distributions. In practice, the GEV distribution is estimated through the block maxima (BM) approach whereas the GP distribution is estimated through the peak over threshold (POT) approach. However, the estimated extreme value distributions are very sensitive to the block size used in the BM approach and to the threshold used in the POT approach. To address these limitations, we introduce the geometric mixture generalized extreme value (GMGEV) distribution. The GMGEV distribution is max- stable, and such, it is a valid asymptotic distribution for maxima. In addition, we propose a statistical modeling approach called the equivalent block maxima (EBM) approach through which the parameters of the GMGEV distribution can be estimated. Specifically, the EBM approach consists to construct the GMGEV distribution with all estimated GEV distributions from the BM approach whose parameters are not statistically different. Illustrations on simulated data show the ability of our extreme value modelling strategy to provide consistent estimators for extreme quantiles. Finally, we apply the proposed strategy to estimate centennial, millennial and deca-millennial return levels associated with river flows and wind speeds in France.