The present study deals with the existence of ‘nearest-neighbour’ type Gibbs models, introduced by Baddeley and Møller in 1989. In such models, the neighbourhood relation depends on the realization of the process. After giving new sufficient conditions to prove the existence of stationary Gibbs states, we deal with the first-nearest-neighbour model, the triplets Delaunay model, Ord's model and Markov connected component type models.

We present an entropy conservation principle applicable to either discrete or continuous variables which provides a useful tool for aggregating observations. The associated method of modality grouping transforms a variable Z1 into a new variable Z2 such that the mutual information I(Z2,Y) between Y, a variable of interest, and Z2 is equal to I(Z1,Y).

A version of the Rao–Blackwell theorem is shown to apply to most, but not all, stereological sampling designs. Estimators based on random test grids typically have larger variance than quadrat estimators; random s-dimensional samples are worse than random r-dimensional samples for s < r. Furthermore, the standard stereological ratio estimators of different dimensions are canonically related to each other by the Rao–Blackwell process. However, there are realistic cases where sampling with a lower-dimensional probe increases efficiency. For example, estimators based on (conditionally) non-randomised test point grids may have smaller variance than quadrat estimators. Relative efficiency is related to issues in geostatistics and the theory of wide-sense stationary random fields. A uniform minimum variance unbiased estimator typically does not exist in our context.

In a counting process considered at time t the focus is often on the length of the current interarrival time, whereas points in the past may be said to constitute information about the process. The paper introduces new concepts on how to quantify predictability of the future behavior of counting processes based on the past information and considers then situations in which the future points become more (or less) predictable. Various properties of our proposed concepts are studied and applications relevant to the reliability of repairable systems are given.