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On a test statistic for linear trend

Journal article, 2004

Consider a change-point detection problem for the appearance of a linear trend in the independent variables $X_i$, where the null hypothesis $H_0$ is that $X_i=e_i$ are standardized discrete white noise, and the alternative is $$ X_i=\cases a_0+a_1(i/n)+e_i,&\text{for} i=1,2,\dots,k,\\ e_i,&\text{for} i=k+1,\dots,n, \endcases $$ for some $k$ and some real $a_0,a_1$. Under $H_0$, the test statistic $$ max_{[\alpha n]\leq k\leq n}\frac{(\sum_1^k X_i)^2}{k}+ \frac{(\sum_1^k((i/n)-(k+1)/2n)X_i)^2} {(\sum_1^k((i/n)-(k+1)/2n))^2} $$ tends in distribution to $\sup_{t\in[\alpha,1]}|Y(t)|^2$ as $n\to\infty$, where $Y(t)$ is a bivariate process defined in terms of a standard Wiener process $W(t)$, $$ Y(t)=\left(\frac {W(t)}{\sqrt{t}},\frac{\sqrt{3}tW(t)- \sqrt{12}\int_0^t W(s)\,ds}{\sqrt{t^3}} \right). $$
In this paper, the asymptotic behaviour of $$P(\sup_{t\in[\alpha,t]}|Y(t)|^2>u)$$ and of $$P(\sup_{t\in[\exp(-e^{u/2}/u),1]}|Y(t)|^2>u+2x)$$ are shown to be $-\ln\alpha$ and $1-\exp(-e^{-x})$, respectively, as $u\to\infty$.