Summary: Characterizations of probability distributions is a topic of great popularity in the applied probability and reliability literature for over the last 30 years. Beside the intrinsic mathematical interest (often related to functional equations) the results in this area are helpful for probabilistic and statistical modelling, especially in engineering and biostatistical problems. A substantial number of characterizations has been devoted to a legion of variants of exponential distributions. The main reliability measures associated with a random vector \(X\) are the conditional moment function defined by \(m_\varphi(x)=E(\varphi(X)\mid X \geq x)\) (which is equivalent to the mean residual life function \(e(x)=m_\varphi (x)-x\) when \(\varphi(x)=x)\) and the hazard gradient function \(h(x)=-\nabla\log R(x)\), where \(R(x)\) is the reliability (survival) function, \(R(x) =\text{Pr}(X\geq x)\), and \(\nabla\) is the operator \(\nabla=(\partial/ \partial x_1,\partial/ \partial x_2,\dots,\partial/\partial x_n)\).
We study the consequences of a linear relationship between the hazard gradient and the conditional moment functions for continuous bivariate and multivariate distributions. We obtain a general characterization result which is then applied to characterize Arnold and Strauss’ bivariate exponential distribution and some related models.