In this correspondence, the comprehensive problem of joint power, rate, and subcarrier allocation have been investigated for enhancing the spectral efficiency of multi-user orthogonal frequency-division multiple access (OFDMA) cognitive radio (CR) networks subject to satisfying total average transmission power and aggregate interference constraints. We propose novel optimal radio resource allocation (RRA) algorithms under different scenarios with deterministic and probabilistic interference violation limits based on a perfect and imperfect availability of cross-link channel state information (CSI). In particular, we propose a probabilistic approach to mitigate the total imposed interference on the primary service under imperfect cross-link CSI. A closed-form mathematical formulation of the cumulative density function (cdf) for the received signal-to-interference-plus-noise ratio (SINR) is formulated to evaluate the resultant average spectral efficiency (ASE). Dual decomposition is utilized to obtain sub-optimal solutions for the non-convex optimization problems. Through simulation results, we investigate the achievable performance and the impact of parameters uncertainty on the overall system performance. Furthermore, we present that the developed RRA algorithms can considerably improve the cognitive performance whilst abide the imposed power constraints. In particular, the performance under imperfect cross-link CSI knowledge for the proposed `probabilistic case' is compared to the conventional scenarios to show the potential gain in employing this scheme.