This paper combines the principles of statistical estimation and hypothesis testing to analyse the impact of parameter estimation on an authentication system based on graphical codes. The studied authentication system uses the fact that a code, once printed, undergoes a stochastic and non invertible al- teration. A statistical test applies a likelihood ratio between the model of the authentic printed and scanned image and the model of the reproduced one, with the particularity here that the later model is unknown. The proposed solution consists in using an optimal estimation of the image model coming from observed fake codes in order to perform the likelihood test. Using a second order expansion, we derive a linear re- lation between the quadratic error of the estimated parame- ters and the probability of type II error. We are then able to formulate analytically and practically the error spread region of the Receiver Operating Characteristic (ROC) curves, and to compute the average authentication performance when the receiver has to estimate the opponent print and scan channel.