<p>The four primary moment-based probability density estimation and associated moment-based parameter estimation methods used in classical statistical inference when the probability density functions for the observed data are unknown or are intractable are briefly described as background for the introduction of an entirely new method. This new method is radically different in approach yet provides a solution that requires essentially the same information as the existing methods: (1) model moments with known dependence on unknown parameters and (2) associated sample moments. However, the new method, unlike the classical method of moments and its generalized counterparts, requires only the solution of simultaneous linear equations. A theoretical comparison between the new and old methods is made, and reference is made to the Author’s earlier work on analytical comparisons with Bayesian parameter estimation and decision for cyclostationary processes. The next step required for finding this new method’s place in practice among present methods of moments is an extensive comparison of the performance of these methods applied to a diverse variety of multivariate data sets. This next step is beyond the scope of this theoretical paper, the purpose of which is to demonstrate for the first time that the method from engineering work on inference for cyclostationary processes in communications systems in the 1970s is a genuine method of moments for multivariate data though this non-obvious equivalence was not originally recognized even by the Author or, apparently, anyone else for half a century.</p>