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On the composite Lognormal–Pareto distribution with uncertain threshold

Abstract Introduced to model heavy-tailed actuarial loss data, the composite Lognormal–Pareto distribution is piecewise built from the Lognormal and Pareto densities defined on mutually disjoint intervals. When estimating its parameters, the main challenge is the estimation of the unknown threshold where the density form changes. Motivated by the uncertainty of this parameter, in this article we consider two different approaches: first, we assume that the threshold is modeled by a random variable, and second, by a fuzzy number. We compare these two approaches especially from the parameters estimation point of view, under the continuity condition. We also study in some detail what happens with the Lognormal–Pareto distribution and with some of its characteristics under the fuzzy threshold assumption.