Sökning: onr:"swepub:oai:gup.ub.gu.se/98075" >
On a test statistic...
On a test statistic for linear trend
-
- Albin, Patrik, 1960 (författare)
- Gothenburg University,Göteborgs universitet,Institutionen för matematisk statistik,Department of Mathematical Statistics,University of Gothenburg,Chalmers tekniska högskola,Chalmers University of Technology
-
(creator_code:org_t)
- 2004
- 2004
- Engelska.
-
Ingår i: Extremes. - 1386-1999 .- 1572-915X. ; 6:3, s. 247-258
- Relaterad länk:
-
https://gup.ub.gu.se...
-
visa fler...
-
https://research.cha...
-
visa färre...
Abstract
Ämnesord
Stäng
- 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 $u\to\infty$.
Ämnesord
- NATURVETENSKAP -- Matematik -- Sannolikhetsteori och statistik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Probability Theory and Statistics (hsv//eng)
Publikations- och innehållstyp
- ref (ämneskategori)
- art (ämneskategori)
Hitta via bibliotek
-
Extremes
(Sök värdpublikationen i LIBRIS)
Till lärosätets databas