Ein asymptotischer Satz über den maximalen Minimalabstand von unabhängigen Zufallsvektoren mit Anwendung auf einen Anpassungstest im R^p und auf der Kugel

Summary

Given i.i.d. random vectorsX ₁,X ₂,...,X _n with continuous densityf(x), taking values in a bounded regionG⊂R ^p (p≧1), let

$$D_n = \mathop {\max }\limits_{i = 1,...,n} \{ f(X_i )\} ^{1/p} \min (\mathop {\min }\limits_{j \ne i} |X_i - X_j | ,|X_i - \partial G|)$$

be their “maximum minimum distance”. Then, under the additional assumptions

$$\begin{gathered} \inf \{ f(x):x \in G\} > 0 \hfill \\ \sup \{ |f(x) - f(y)| : |x - y|< \rho \} = o(( - log \rho )^{ - 1} ) (\rho \downarrow 0), \hfill \\ \end{gathered} $$

the asymptotic theorem

$$\mathop {\lim }\limits_{n \to \infty } P\left( {n \cdot \frac{{\pi ^{p/2} }}{{\Gamma (p/2 + 1)}} \cdot D_n^p - \log n< \xi } \right) = \exp ( - e^{ - \xi } ) (\xi \in R^1 )$$

holds. As a statistical application, one readily obtains a test of goodness of fit based onD _n . The limiting power of this test with respect to approaching alternatives is derived. Analogous results apply to the case of random vectors on the surface of a sphere or even more general differentiable structures.