On a Test Statistic for Linear Trend

摘要

Let {W(s)}_(s≥0) be a standard Wiener process. The supremum of the squared Euclidian norm |Y(t)|~2, of the R~2-valued process Y(t) = ((1/t)~(1/2)W(t), (12/t~3)~(1/2) ∫_0~t s dW(s) - (3/t)~(1/2)W(t)), t ∈ [α, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P{sup_(t ∈ [α, 1] |Y(t)|~2 > u} as u → ∞, of this statistic, for a fixed α ∈ (0, 1), and for a "moving" α = α(u) ↓ 0 at a suitable rate as u → ∞. The statistical interest of our results lie in their use as approximate test levels.