Properties of the random effect transformation (Q6592723)

Given the survival function \(\overline{F}(x)=1-F(x)\) of a random variable \(X\), and another random variable \(Z\), the authors consider the distribution function defined by \(F^{(Z)}(x)=1-\mathbf{E}[\overline{F}(x)^Z]\). In particular they show that if \(F\) is regularly varying of index \(\alpha\) then \(F^{(Z)}\) is regularly varying with index \(\alpha b\), where \(b\) is the infimum of the support of \(Z\). The authors further show that if \(F\) is dominatedly varying, long-tailed or generalised long-tailed, then so is \(F^{(Z)}\). Several examples are considered to illustrate these results.