| The observation that the distribution of a complex statistical model can be characterized as the stationary distribution of a Markov process plays a central role in a variety of statistical and probabilistic methods. This characterization defines Markov chain Monte Carlo methods, and it determines the family of unbiased estimating equations in Baddeley's time-invariance estimation method. Typically, in these methods, one begins with the model of interest specified in some way and then constructs a Markov process having the desired distribution as its stationary distribution; however, one can also specify the Markov process and then take the model to be the corresponding stationary distribution. This reverse approach is flexible (rich parametric families of models can easily be specified) and intuitive (qualitative properties of the distribution can usually be inferred from the dynamic properties of the Markov process). This reverse approach of modeling way is used in this paper. The parameter estimates are obtained by equating to zero the generator of Y applied to a collection of suitable statistics. However, Baddeley does not give any suggestions for these statistics. We suggest some statistics for largely two-parameter spatial point processes including Gibbs and non-Gibbs models. Moreover, under the selections we suggest, consistency of the time-invariance estimates can be proved. Moreover, under the selections we suggest, the results of simulations are provided and some examples for real data sets are analyzed. |