On sums of dependent random lifetimes under the time-transformed exponential model

For a given pair of random lifetimes whose dependence is described by a time-transformed exponential model, we provide analytical expressions for the distribution of their sum. These expressions are obtained by using a representation of the joint distribution in terms of bivariate distortions, which is an alternative approach to the classical copula representation. Since this approach allows one to obtain conditional distributions and their inverses in simple form, then it is also shown how it can be used to predict the value of the sum from the value of one of the variables (or vice versa) by using quantile regression techniques.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. JN and JM are supported by Ministerio de Ciencia e Innovación of Spain under Grant PID2019-103971GB-I00/AEI/10.13039/501100011033. FP is partially supported by the Grant Progetto di Eccellenza, CUP: E11G18000350001 and by the Italian GNAMPA research group of INdAM (Istituto Nazionale di Alta Matematica)