Massive multiple-input multiple-output (MIMO) promises significantly higher performance relative to conventional multiuser systems. However, the promised gains of massive MIMO systems rely heavily on the accuracy of the absolute front-end calibration, as well as quality of channel estimates at the base station (BS). In this paper, we analyze user equipment-aided calibration mechanism to estimate the amplitude scaling and phase drift at each radio-frequency chain interfacing with the BS array. Assuming a uniform linear array at the BS and Ricean fading, we obtain the estimation parameters with moment-based (amplitude, phase) and maximum-likelihood (phase-only) estimation techniques. In stark contrast to previous works, we mathematically articulate the equivalence of the two approaches for phase estimation. Furthermore, we rigorously derive a Cramer-Rao lower bound to characterize the accuracy of the two estimators. Via numerical simulations, we evaluate the estimator performance with varying dominant line-of-sight powers, dominant angles-of-arrival, and signal-to-noise ratios.