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In previous work, it has been shown that the uplink feedback load required for downlink scheduling can be reduced when a spread-spectrum contention feedback channel is employed. Particularly, the feedback load can be reduced if only the users whose downlink rates are larger than a prespecified threshold are allowed to report their downlink rates through the contention channel. Thus, it is crucial to determine the access threshold for each user, which requires the exact distribution of each user's maximum achievable downlink rate. However, the rate distribution is usually unknown. In this paper, each user finds the largest maximum achievable downlink rate by selecting the best transmit antenna out of multiple downlink transmit antennas and stores the rate in the set of the most recently measured maximum achievable rates. These measured rates are used to determine the threshold to be used for scheduling. The stored rates are placed in descending order, and the threshold is determined as the rate occupying a particular position in the ordered rates. When the maximum achievable rate is larger than the threshold, the user sends the rate and the corresponding antenna to the base station through the contention channel. Under this setup, we derive two different lower bounds on the average throughput ratio of the proposed scheme and the scheme with the exact distribution. In particular, they are a universal lower bound that is valid for any maximum achievable rate distribution and a second lower bound that is an approximated lower bound that is tighter than the universal lower bound while only requiring the information on the measured maximum achievable rates. Using these results, it is shown that the average user throughput of the proposed scheme is always smaller than the average user throughput of the exact-distribution-based scheme and exponentially approaches that of the exact scheme as the number of stored rates used to determine the threshold increases.