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In this paper, performance of cognitive transmission over time-selective flat fading channels is studied under quality of service (QoS) constraints and channel uncertainty. Cognitive secondary users (SUs) are assumed to initially perform channel sensing to detect the activities of the primary users, and then attempt to estimate the channel fading coefficients through training. Energy detection is employed for channel sensing, and different minimum mean-square-error (MMSE) estimation methods are considered for channel estimation. In both channel sensing and estimation, erroneous decisions can be made, and hence, channel uncertainty is not completely eliminated. In this setting, performance is studied and interactions between channel sensing and estimation are investigated. Following the channel sensing and estimation tasks, SUs engage in data transmission. Transmitter, being unaware of the channel fading coefficients, is assumed to send the data at fixed power and rate levels that depend on the channel sensing results. Under these assumptions, a state-transition model is constructed by considering the reliability of the transmissions, channel sensing decisions and their correctness, and the evolution of primary user activity which is modeled as a two-state Markov process. In the data transmission phase, an average power constraint on the secondary users is considered to limit the interference to the primary users, and statistical limitations on the buffer lengths are imposed to take into account the QoS constraints of the secondary traffic. 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.