Testing for epidemic changes in the mean of a multiparameter stochastic process

• We model random fields with a possible structural break over a rectangle.
• We propose a consistent test for a change in the mean of a random field.
• Using the theory of extremes of Gaussian fields, we derive approximate critical values.
• Under the alternative, we provide an estimator for the rectangle with changed mean.

In this paper, multiparameter stochastic processes {Zn(x)}x∈[0,n]d{Zn(x)}x∈[0,n]d, n∈Nn∈N, are tested for the occurrence of a block (k0,m0]=(k0,1,m0,1]×…×(k0,d,m0,d]⊂[0,n]d(k0,m0]=(k0,1,m0,1]×…×(k0,d,m0,d]⊂[0,n]d where the mean of the process changes. This is modeled in the formZn(x)=λ((0̲,x])μn+σY(x)+λ((0̲,x]∩(k0,m0])δn,where 0̲=(0,…,0)′, λ(A  ) denotes the Lebesgue measure of a set A⊂RdA⊂Rd, and μnμn, δn∈Rδn∈R as well as 0<σ<∞0<σ<∞ are unknown parameters. The stochastic process {Yn(t)=Y(⌊nt⌋):t∈[0,1]d} is assumed to fulfill a weak invariance principle. Under the null hypothesis, an approximation for the tail behavior of the limit variable of a trimmed maximum-type test statistic is given. Then, the (weak) consistency of the test under the alternative is proven. The corresponding estimation problem for the points k0k0 and m0m0 is also considered and consistent estimators are given for local alternatives δn=δn−d/2δn=δn−d/2 with δ≠0δ≠0.