<p>The five-parameter bivariate Birnbaum-Saunders and bivariate lognormal distributions are two models used to study positive lifetime bivariate data. These distributions exhibit striking similarities, from the surface plot of their joint density functions to the nature of their reliability functions and hazard gradients within certain parameter spaces. Additionally, their marginal distributions share similar shapes. This paper aims to discriminate between these two distributions. To achieve this, we use the difference between the maximized log-likelihood functions as a discrimination statistic. The asymptotic distribution of the test statistic is derived to compute the probability of correct selection (PCS). Moreover, we have also studied the effects of model misspecification on the mode of the marginal hazard function as well as on the components of the hazard gradient. Due to the limitations of the usual likelihood approach, we also propose a suitable modification for the decision rule. Finally, we do a real data analysis.</p>