Peter Dencker
A Limit Theorem for empirical processes from planned experiments
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 52, 65-84 (1998)

MSC:

60G60 Random fields

62G20 Asymptotic properties

Abstract: In this paper the data $Y_1,\ldots,Y_m$ have the following structure. The
measurements are taken at $t_i\in (0,1)^q$ and there is a family of
distributions $Q_{\eta},\,\eta\in(a,b)$ and $g_0:\,[0,1]^q\to (a,b)$ such
that $\mathcal{L}(Y_i)=Q_{g_0(t_i)}$. The main results of this paper is a
limit theorem for the sequence of random fields
\[ W_m(t)=\frac{1}{\sqrt{m}}\Sum_{j=1}^{m}
\left(Y_j-g_0(t_j)\right) I_{(0,t]}(t_j)\ .\]
Limit theorems for functionals of $W_m$ are used to construct an asymptotic
$\alpha$-test for $H_0:\,g=g_0$ versus $H_A:\,g\not=g_0$. For a regression
model the asymptotic power is investigated under local alternatives.
Keywords: Random fields, Limit Theorems, Wiener field, Goodness of fit test