Several studies showed that EEG signal of Alzheimer's disease patients is less complex than that of healthy subjects. In this article, we propose to characterize the complexity of the EEG signal by an entropy measure based on local density estimation by a Hidden Markov Model. We first show that this measure leads to consistent results qualitatively and quantitatively (in terms of classification accuracy). Indeed, it discriminates AD patients, at an early stage of Alzheimer's disease, from healthy subjects: a classification accuracy of 80% is reached on a dataset including EEG data recorded in different conditions. Based on this measure, we also show that parietal and temporal regions are the first regions affected by complexity loss in the early stage of Alzheimer's disease.