  Since the two seminal papers by Fisher (1915, 1921) were published, the test
under a fixed value correlation coefficient null hypothesis for the bivariate
normal distribution constitutes an important statistical problem. In the
framework of asymptotic robust statistics, it remains being a topic of great
interest to be investigated. For this and other tests, focused on paired
correlated normal random samples, R\'{e}nyi's pseudodistance estimators are
proposed, their asymptotic distribution is established and an iterative
algorithm is provided for their computation. From them the Wald-type test
statistics are constructed for different problems of interest and their
influence function is theoretically studied. For testing null correlation in
different contexts, an extensive simulation study and two real data based
examples support the robust properties of our proposal.
