Effective sample size of spatial process models

This paper focuses on the reduction of sample sizes due to the effect of autocorrelation for the most common models used in spatial statistics. This work is an extension of a simple illustration highlighted in several books for an autoregressive-type correlation structure. The paper briefly reviews existing proposals to quantify the effective sample size and proposes a new definition that is a function of the correlation structure, sample size, and dimension of the space where the coordinates are defined. It describes the properties of and explicit expression for the effective sample size for processes with patterned correlation matrices, including elliptical contoured distributions. The estimation of the effective sample size is achieved using restricted maximum likelihood. Additionally, the paper describes the monotonicity of the effective sample size when two random points are uniformly distributed on the unit sphere and includes several Monte Carlo simulations to explore monotonic features of the effective sample size and to compare its behavior with respect to other proposals. Finally, this paper analyzes two real datasets, and the discussion includes topics that should be addressed in further research.