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.