This article first demonstrates the inconsistency of the estimator based on the standard Allan Variance (AV) for composite stochastic processes. This result motivates the use of a recently developed estimator, called the Generalized Method of Wavelet Moments (GMWM) estimator. This estimator was previously shown to be consistent and asymptotically normally distributed under the settings of the present research. Moreover, and unlike Maximum Likelihood Estimators (MLE), this method is able to estimate parameters of complex composite stochastic processes. After establishing the link between the AV and GMWM estimators we prove that there always exists a GMWM estimator with smaller asymptotic variance than its AV based counterpart. As the GMWM may be biased in finite samples issued from simple models on which the AV or MLE can still be applied, we present several extensions to the GMWM which significantly enhance its performance. One of these extensions is a finite sample bias correction related to the principle of indirect inference which is very general and can be applied beyond the scope of the GMWM framework. Finally, the theoretical findings are supported by simulation studies that compares the finite sample performance of AV based estimators, MLE and the various different forms of GMWM estimators.