A class of U-statistics and asymptotic normality of the number of k-clusters

Rabi N. Bhattacharya; Jayanta K. Ghosh

doi:10.1016/0047-259X(92)90038-H

A class of U-statistics and asymptotic normality of the number of k-clusters

Rabi N. Bhattacharya, Jayanta K. Ghosh

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

A central limit theorem is proved for a class of U-statistics whose kernel depends on the sample size and for which the projection method may fail, since several terms in the Hoeffding decomposition contribute to the limiting variance. As an application we derive the asymptotic normality of the number of Poisson k-clusters in a cube of increasing size in R^d. We also extend earlier results of Jammalamadaka and Janson to general kernels and to general orders k > 2 of the kernel.

Original language	English (US)
Pages (from-to)	300-330
Number of pages	31
Journal	Journal of Multivariate Analysis
Volume	43
Issue number	2
DOIs	https://doi.org/10.1016/0047-259X(92)90038-H
State	Published - Nov 1992
Externally published	Yes

Keywords

Poisson random field
U-statistics
interpoint distance
k-clusters
martingales

ASJC Scopus subject areas

Statistics and Probability
Numerical Analysis
Statistics, Probability and Uncertainty

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10.1016/0047-259X(92)90038-H

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AU - Bhattacharya, Rabi N.

AU - Ghosh, Jayanta K.

PY - 1992/11

Y1 - 1992/11

N2 - A central limit theorem is proved for a class of U-statistics whose kernel depends on the sample size and for which the projection method may fail, since several terms in the Hoeffding decomposition contribute to the limiting variance. As an application we derive the asymptotic normality of the number of Poisson k-clusters in a cube of increasing size in Rd. We also extend earlier results of Jammalamadaka and Janson to general kernels and to general orders k > 2 of the kernel.

AB - A central limit theorem is proved for a class of U-statistics whose kernel depends on the sample size and for which the projection method may fail, since several terms in the Hoeffding decomposition contribute to the limiting variance. As an application we derive the asymptotic normality of the number of Poisson k-clusters in a cube of increasing size in Rd. We also extend earlier results of Jammalamadaka and Janson to general kernels and to general orders k > 2 of the kernel.

KW - Poisson random field

KW - U-statistics

KW - interpoint distance

KW - k-clusters

KW - martingales

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VL - 43

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EP - 330

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 2

ER -

A class of U-statistics and asymptotic normality of the number of k-clusters

Abstract

Keywords

ASJC Scopus subject areas

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