Pseudo-operations and pseudo-analysis: applications to probability and risk modelling

The aim of this thesis is to generalize some well-known concepts in probability and statistics from the standard ring to the non-idempotent semi-ring (R+, ⊕h, ⊗h), where pseudo-sum ⊕h and ⊗h are defined by a suitable function h, called generator. In the first part of this dissertation, we introduce the notions of pseudo-independence with respect to a pseudo-additive fuzzy measure and of pseudo-moment generating functions, showing that the classical results concerning moment generating functions of a vector of independent random variables and of their sum extend to pseudo-moment generating functions if the random variables involved are pseudo-independent. Moreover, we prove that pseudo moment generating functions, and more in general pseudo-analysis, can be particularly efficient in characterizing a new class of bivariate random vectors that we call ”pseudo-Schur constant” family, which represents an extension of the well-known Schur-constant class. In the second part of this thesis, we give a generalization of strong and weak bivariate lack-of-memory properties, substituting into their associated functional equations the standard product by the pseudo one ⊗h: we call the distributions satisfying them pseudo strong and weak distributions. After characterising the pseudo weak distribution in full generality, we study the induced dependence structure of the underlying lifetimes and that of the residual ones. Moreover, we show that the distributions satisfying pseudo lack-of-memory properties coincide with the solutions of suitable generalizations of Kaminsky (1983) and Marshall and Olkin (2015) functional equations; finally, we analyse several examples of pseudo weak distributions that may be used in life insurance and we give a non-life insurance application to LOSS and ALAE modelling problem.