Saddlepoint-based bootstrap inference for the spatial dependence parameter in the lattice process

A saddlepoint-based bootstrap (SPBB) method for inference on the spatial dependence parameter in Gaussian lattice regression models is proposed. The method handles any estimator that is expressible as the root of a quadratic estimating equation (QEE), and includes common estimators like maximum likelihood (ML) and restricted ML (REML). Since the underlying QEE has a moment generating function in closed-form, this is inverted via the saddlepoint method to produce accurate approximations to the respective distributions of the estimators. Confidence intervals are then produced by pivoting the distribution function. The approach provides a unified perspective by viewing estimators in the simultaneous autoregressive (SAR), conditional autoregressive (CAR), and simultaneous moving average (SMA) models, through their underlying QEEs. The key assumption of monotonicity for the respective QEEs is verified, and results derived concerning their bias. Of importance for spatial modeling practitioners, is the finding that simulation studies show SPBB confidence intervals outperforming those based on standard (first-order) asymptotic theory in small to moderate sample size settings, and being orders of magnitude faster than computationally intensive approaches like the bootstrap. The methodology is illustrated on two classical lattice datasets.