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In this paper, we propose a comprehensive framework to jointly analyze the angle estimation error and design the three-dimensional (3D) positioning algorithm for a millimeter wave (mmWave) positioning system. First, we estimate the angles of arrival (AoAs) at the anchors by applying the two-dimensional discrete Fourier transform (2D-DFT) algorithm. Based on the property of the 2D-DFT algorithm, the angle estimation error is analyzed in terms of probability density functions (PDF). The analysis shows that the derived angle estimation error is non-Gaussian, which is different from the existing work. Second, the intricate expression of the PDF of the AoA estimation error is simplified by employing the first-order linear approximation of triangle functions. Then, we derive a complex expression for the variance based on the derived PDF. Specifically, for the azimuth estimation error, the variance is separately integrated according to the different non-zero intervals of the PDF. Finally, we apply the weighted least square (WLS) algorithm to estimate the 3D position of the MU by using the estimated AoAs and the obtained non-Gaussian variance. Extensive simulation results confirm that the derived angle estimation error is non-Gaussian, and also demonstrate the superiority of the proposed framework.