In this paper, performance of transmission in cognitive radio systems over time-selective flat fading channels is studied under quality of service (QoS) constraints and channel uncertainty. Cognitive secondary users are assumed to initially perform sensing over the transmission channel to detect the activities of the primary users. Then, depending on the channel sensing result, they choose their transmission power policies and perform channel estimation. Following the training phase, they transmit data through the channel. The activities of the primary users are modeled as a two-state Markov process. A state transition model is constructed to model the cognitive transmissions. Statistical limitations on the buffer lengths are imposed to take into account the QoS constraints, and an average power constraint on the secondary users is considered to limit the interference to the primary users. The maximum throughput under these statistical QoS constraints is identified by finding the effective capacity of the cognitive radio channel. Numerical results are provided for the power and rate policies.