Resonators are passive devices that respond to an excitation signal by oscillating at their natural frequency with exponentially decreasing amplitudes. Physical, chemical and electrical variables can modify the natural frequencies of resonators. If resonators are connected to antennas or other transducers that couple into a communication channel, they enable purely passive sensors that can be read wirelessly. In this manuscript, we use maximum likelihood estimation to analyze the measurement accuracy that can be achieved by the wireless readout of passive resonant sensors as a function of the read signal, the stimulation power and noise figure of the reader, the distance and transducer gain of the transmission channel, and the natural frequency and quality factor of the resonant passive sensor. The Crámer-Rao lower bound characterizes the minimum variance of the natural frequency and decay constant of the resonator. We show the derivation of the Crámer-Rao lower bounds from the Fisher information matrix based on a maximum likelihood estimation of discrete-time samples of an exponentially decaying phasor. This theoretical lower limit of accuracy is almost achieved by an iterative algorithm that approximates the maximum of the measured resonator spectrum with a Lorentz curve.