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In this paper, we will derive closed-form expressions of false alarm probabilities for a given threshold for the dimension estimation-based detector (DED) using Akaike information criterion (AIC) and the minimum description length (MDL) criterion. Specifically, the DED algorithm will be formulated as a binary hypothesis test using AIC and MDL curves. Based on the proposed statistic test, we will express the probability of false alarm of the DED algorithm for a fixed threshold using the cumulative density function (CDF) for the distribution of Tracy-Widom of order two. The derived analytical decision thresholds are verified with Monte-Carlo simulations and a comparison between simulation and analytical results to confirm the theoretical results are presented. These results confirm the very good match between simulation and theoretic results.