Consistent parametric estimation of the intensity of a spatial-temporal point process

We consider conditions under which parametric estimates of the intensity of a spatial-temporal point process are consistent. Although the actual point process being estimated may not be Poisson, an estimate involving maximizing a function that corresponds exactly to the log-likelihood if the process is Poisson is consistent under certain simple conditions. A second estimate based on weighted least squares is also shown to be consistent under quite similar assumptions. The conditions for consistency are simple and easily verified, and examples are provided to illustrate the extent to which consistent estimation may be achieved. An important special case is when the point processes being estimated are in fact Poisson, though other important examples are explored as well.