Long-run variance estimation for spatial data under change-point alternatives

•We model multivariate random fields with a possible structural break.•We propose a long-run variance matrix (LRV) estimator that takes possible breaks in the mean into account.•We study the asymptotic behavior of the proposed LRV estimator and give conditions for its consistency.•Under the assumption of a change over a single rectangle, we provide a consistent estimator for the rectangle with changed mean and give a rate for its convergence.

In this paper, we consider the problem of estimating the long-run variance (matrix) of an RpRp-valued multiparameter stochastic process {Xk}k∈[1,n]d, (n,p,d∈Nn,p,d∈N,p,dp,d fixed) whose mean-function has an abrupt jump. We consider processes of the form Xk=Yk+μ+ICn(k)Δ, where ICIC is the indicator function for a set CC, the change-set Cn⊂[1,n]dCn⊂[1,n]d is a finite union of rectangles and μ,Δ∈Rpμ,Δ∈Rp are unknown parameters. The stochastic process {Yk:k∈Zd} is assumed to fulfill a weak invariance principle. Due to the non-constant mean, kernel-type long-run variance estimators using the arithmetic mean of the observations as a mean estimator have an unbounded error for changes ΔΔ that do not vanish for n→∞n→∞. To reduce this effect, we use a mean estimator which is based on an estimation of the set CnCn. In the case where Cn=(⌊nθ10⌋,⌊nθ20⌋] is a rectangle, we introduce an estimator Cˆn=(⌊nθˆ1⌋,⌊nθˆ2⌋] and study its convergence rate.