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A correct modelization of the insurance losses distribution is crucial in the insurance industry. This distribution is generally highly positively skewed, unimodal hump-shaped, and with a heavy right tail. Compound models are a profitable way to accommodate situations in which some of the probability masses are shifted to the tails of the distribution. Therefore, in this work, a general approach to compound unimodal hump-shaped distributions with a mixing dichotomous distribution is introduced. A 2-parameter unimodal hump-shaped distribution, defined on a positive support, is considered and reparametrized with respect to the mode and to another parameter related to the distribution variability. The compound is performed by scaling the latter parameter by means of a dichotomous mixing distribution that governs the tail behavior of the resulting model. The proposed model can also allow for automatic detection of typical and atypical losses via a simple procedure based on maximum a posteriori probabilities. Unimodal gamma and log-normal are considered as examples of unimodal hump-shaped distributions. The resulting models are firstly evaluated in a sensitivity study and then fitted to two real insurance loss datasets, along with several well-known competitors. Likelihood-based information criteria and risk measures are used to compare the models. [ABSTRACT FROM AUTHOR]

Portmanteau tests are typically used to test serial independence even if, by construction, they are generally powerful only in presence of pairwise dependence between lagged variables. In this article, we present a simple statistic defining a new serial independence test, which is able to detect more general forms of dependence. In particular, differently from the Portmanteau tests, the resulting test is powerful also under a dependent process characterized by pairwise independence. A diagram, based on p-values from the proposed test, is introduced to investigate serial dependence. Finally, the effectiveness of the proposal is evaluated in a simulation study and with an application on financial data. Both show that the new test, used in synergy with the existing ones, helps in the identification of the true data-generating process. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]