In Chapter 6, a set of special sub-families of annual loss loss distribution approach (LDA) models were developed which admitted closed-form exact representations. That is, closed-form representations were obtained for the distribution of the annual loss *Z* = *X*_{1} + · · · + *X*_{N} represented by a compound process model with LDA structure in which the frequency is *N* ~ *Poisson*(*λ*) and the severities are independent and identically distributed (i.i.d.) *X*_{i}(*t*) ~ *F*_{X}(*x*), and *N* and *X*_{i} are independent. The exact distribution of the annual loss processes can be expressed analytically as a mixture distribution comprised convolved distributional components with Poisson mixing weights for *N* > 0,

Here, = r[*X*_{1} + *X*_{2} + · · · + *X*_{n} ≤ *z*] is the *n*-th convolution of the severity distribution *F*_{X}(·) calculated recursively as

with = 1 if *z* ≥ 0 and zero otherwise. Note, throughout this chapter we refer to *F* as the distribution for the severity, ...

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