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

In this paper, multiparameter stochastic processes { Z n ( x ) } x ∈ [ 0 , n ] d , n ∈ N , are tested for the occurrence of a block ( k 0 , m 0 ] = ( k 0 , 1 , m 0 , 1 ] × … × ( k 0 , d , m 0 , d ] ⊂ [ 0 , n ] d where the mean of the process changes. This is modeled in the form x ) = λ ( ( 0 ̲ , x ] ) μ n + σ Y ( x ) + λ ( ( 0 ̲ , x ] ∩ ( k 0 , m 0 ] ) δ n , where 0 ̲ = ( 0 , … , 0 ) ′ , λ(A) denotes the Lebesgue measure of a set A ⊂ R d , and μ n , δ n ∈ R as well as 0 < σ < ∞ are unknown parameters. The stochastic process { Y n ( t ) = Y ( ⌊ n t ⌋ ) : 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 k 0 and m 0 is also considered and consistent estimators are given for local alternatives δ n = δ n − d / 2 with δ ≠ 0 .