Ryu-type extended Marshall-Olkin model with implicit shocks and joint life insurance applications

In this paper we suggest an improvement of the Extended Marshall-Olkin methodology by allowing an implicit effect of the common shocks affecting the elements of the system. Properties of this new model are studied. We propose an empirical application to a sample of censored residual lifetimes of couples of insureds extracted from a data set of annuities contracts of a large Canadian life insurance company. We obtain estimation of the model parameters using a two-stage maximum likelihood technique and discuss the obtained results.©2021 Elsevier B.V. All rights reserved.1. Introduction and preliminariesThe classical bivariate Marshall-Olkin (MO) shock model has a long history since the seminal paper of Marshall and Olkin (1967). It is specified by the stochastic representation(X1,X2)=(min(T1,T3),min(T2,T3)),(1)where non-negative continuous random variables T1and T2identify the occurrence of independent “individual shocks” affecting two devices and T3is their “common shock” arrival time under the assumption that the shocks are governed by independent homogeneous Poisson processes, i.e., Ti’s in (1 )are exponentially distributed. The random vector (X1, X2)represents the joint distribution of both lifetimes and let us denote its joint survival function by SX1,X2(x1,x2)=P(X1>x1,X2>x2)for all x1, x2≥0.In general, the MO construction (1)implies that the distribution of (X1, X2)has a singularity along the line {x1=x2}generated by the occurrence of the simultaneous default of both elements in the system, due to the fact that P(X1=X2) >0.The stochastic relation (1)can be equivalently rewritten asSX1,X2(x1+t,x2+t)=SX1,X2(x1,x2)SX1,X2(t,t)for allx1,x2,t≥0,(2)characterizing the bivariate lack of memory property (BLMP). The only solution with exponential marginals of the functional equation (2)is given by*Corresponding author.E-mail address:sabrina.mulinacci@unibo.it(S. Mulinacci).https://doi.org/10.1016/j.insmatheco.2021.08.0070167-6687/©2021 Elsevier B.V. All rights reserved.