Abstract
In practical data analysis, methods based on proximity (near-neighbour) relationships between sample points are important because these relations can be computed in time ( n log n ) as the number of points n →∞. Associated with such methods are a class of random variables defined to be functions of a given point and its nearest neighbours in the sample. If the sample points are independent and identically distributed, the associated random variables will also be identically distributed but not independent. Despite this, we show that random variables of this type satisfy a strong law of large numbers, in the sense that their sample means converge to their expected values almost surely as the number of sample points n →∞.
Highlights
Let X ZX1, X2, . be a sequence of independent and identically distributed random vectors Xi 2 Rd, and let X n Z ðX1; .; XnÞ denote the first n points of the sequence
For a sequence X ZX1, X2, . of independent and identically distributed random vectors in Rd, we have proved a strong law of large numbers for functions of a point and its nearest neighbours in the sample X n Z ðX1; .; XnÞ
We have used the result to show that certain non-parametric difference-based estimators of residual moments are strongly consistent as the number of points n/N
Summary
Let X ZX1, X2, . be a sequence of independent and identically distributed random vectors Xi 2 Rd, and let X n Z ðX1; .; XnÞ denote the first n points of the sequence. Motivated by the need for a multidimensional goodness-of-fit test, Bickel & Breiman (1983) investigated the asymptotic properties of sums of bounded functions of the (first) nearest neighbour distances, and proved a central limit theorem for the random variables f ðXiÞkXi K Xniðn;1Þk where f is the (unknown) sampling density. This was extended to k-nearest neighbour distances by Penrose (2000), with k being allowed to increase as a fractional power of the number of points n. The sample mean Gn is a strongly consistent estimator for the kth moment E(Rk ) of the residual distribution as n/N
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