Recent results suggest that measurements of the average “predominant period,” τp, of an earthquake using the first few seconds of P waves provide a rapid approximate estimate of magnitude that can be used for earthquake early warning systems. Although these prior studies demonstrate the empirical value of such an approach, we here examine the theoretical properties of the predominant period estimator. We show that this estimator is a nonlinear function of spectral amplitude and period that gives greater weight to higher amplitudes and higher frequencies in the spectrum. Our results also demonstrate that there are inherent errors in individual measurements using a time-dependent maximum value of the τp estimator derived from a recursion relation. Whereas some of the observed variability in predominant period estimates of magnitude likely is due to local site effects, our analyses suggest that the nonideal properties of the estimator may also add noise to the results. Given the potential importance for earthquake early warning systems, we suggest that more detailed analyses into the magnitude dependence of the spectral characteristics of initial P-wave data, such as using multitaper spectral or wavelet approaches, could be helpful in designing improved methods to further optimize performance.