The complete part of the earthquake frequency-magnitude distribution (FMD), above completeness magnitude mc, is well described by the Gutenberg-Richter law. The parameter mc however varies in space due to the seismic network configuration, yielding a convoluted FMD shape below max(mc). This paper investigates the shape of the generalized FMD (GFMD), which may be described as a mixture of elemental FMDs (eFMDs) defined as asymmetric Laplace distributions of mode mc [Mignan, 2012, https://doi.org/10.1029/2012JB009347]. An asymmetric Laplace mixture model (GFMD- ALMM) is thus proposed with its parameters (detection parameter kappa, Gutenberg-Richter beta-value, mc distribution, as well as number K and weight w of eFMD components) estimated using a semi-supervised hard expectation maximization approach including BIC penalties for model complexity. The performance of the proposed method is analysed, with encouraging results obtained: kappa, beta, and the mc distribution range are retrieved for different GFMD shapes in simulations, as well as in regional catalogues (southern and northern California, Nevada, Taiwan, France), in a global catalogue, and in an aftershock sequence (Christchurch, New Zealand). We find max(mc) to be conservative compared to other methods, kappa = k/log(10) = 3 in most catalogues (compared to beta = b/log(10) = 1), but also that biases in kappa and beta may occur when rounding errors are present below completeness. The GFMD-ALMM, by modelling different FMD shapes in an autonomous manner, opens the door to new statistical analyses in the realm of incomplete seismicity data, which could in theory improve earthquake forecasting by considering c. ten times more events.
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