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Scaling processes of parameter α are ubiquitous in science and engineering. Depending upon α, stationary and nonstationary models are obtained. The presence or absence of stationarity dictates the choice of the analysis methods, estimation techniques and stochastic models to be used. Wavelet entropy has recently been proposed as a powerful tool to describe the degree of order/disorder in a time series. This paper generalizes Shannon wavelet entropy and based on the study of entropy planes and filtering properties, proposes the use of Tsallis entropies of order β to effectively discriminate between scaling processes of parameter α in the vicinity of α <; 1 - |ϵ|, ϵ ∈ R, ϵ ∈, (0.1, 0.5). The influence of β in the discrimination process is discussed in some detail. Theoretical results are validated by experimental studies where numerous fBM and fGn signals were artificially generated.