Nonparametric partitioning estimation of residual and local variance based on first and second nearest neighbours

P. G. Ferrario and H. Walk

Journal of Nonparametric Statistics, 2012, vol. 24, issue 4, 1019-1039

Abstract: In this paper, we consider first an estimator of the residual variance treated by Evans [(2005), 'Estimating the Variance of Multiplicative Noise', in 18th International Conference on Noise and Fluctuations, ICNF , in AIP Conference Proceedings , 780, pp. 99-102], Evans and Jones [(2008), 'Non-Parametric Estimation of Residual Moments and Covariance', Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 464, 2831-2846] and by Liitiäinen, Corona, and Lendasse [(2008), 'On Nonparametric Residual Variance Estimation', Neural Processing Letters , 28, 155-167; (2010), 'Residual Variance Estimation Using a Nearest Neighbour Statistic', Journal of Multivariate Analysis , 101, 811-823], based on first and second nearest neighbours given an independent and identically distributed sample. Its strong consistency and almost sure convergence of the arithmetic means sequence are shown under mere boundedness and square integrability, respectively, of the response variable Y . Moreover, in view of the local variance, a correspondingly modified estimator of local averaging (partitioning) type is proposed, and strong L 2 -consistency for bounded Y , weak L 2 -consistency and optimal rate of convergence (for bounded X under suitable Hölder continuity conditions on regression and local variance functions) under moment conditions on Y are established. Simulation studies illustrate the behaviour of the local variance estimates.

Date: 2012
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DOI: 10.1080/10485252.2012.716836

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