Mathematics – Statistics Theory
Scientific paper
2007-11-28
Bernoulli 2007, Vol. 13, No. 4, 1124-1150
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.3150/07-BEJ5167 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statist
Scientific paper
10.3150/07-BEJ5167
Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed with a common density, define the measure $\nu_n=\sum_i\xi_i$, where $\xi_i$ is a measure (not necessarily a point measure) which stabilizes; this means that $\xi_i$ is determined by the (suitably rescaled) set of points near $X_i$. For bounded test functions $f$ on $R^d$, we give weak and strong laws of large numbers for $\nu_n(f)$. The general results are applied to demonstrate that an unknown set $A$ in $d$-space can be consistently estimated, given data on which of the points $X_i$ lie in $A$, by the corresponding union of Voronoi cells, answering a question raised by Khmaladze and Toronjadze. Further applications are given concerning the Gamma statistic for estimating the variance in nonparametric regression.
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