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

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  • P. G. Ferrario
  • H. Walk

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.

Suggested Citation

  • P. G. Ferrario & H. Walk, 2012. "Nonparametric partitioning estimation of residual and local variance based on first and second nearest neighbours," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(4), pages 1019-1039, December.
    • Handle: RePEc:taf:gnstxx:v:24:y:2012:i:4:p:1019-1039
    DOI: 10.1080/10485252.2012.716836
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    References listed on IDEAS

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    1. Cai, T. Tony & Levine, Michael & Wang, Lie, 2009. "Variance function estimation in multivariate nonparametric regression with fixed design," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 126-136, January.
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    3. Liitiäinen, Elia & Corona, Francesco & Lendasse, Amaury, 2010. "Residual variance estimation using a nearest neighbor statistic," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 811-823, April.
    4. Devroye Luc & Schäfer Dominik & Györfi László & Walk Harro, 2003. "The estimation problem of minimum mean squared error," Statistics & Risk Modeling, De Gruyter, vol. 21(1/2003), pages 15-28, January.
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    6. Fan, Jianqing & Yao, Qiwei, 1998. "Efficient estimation of conditional variance functions in stochastic regression," LSE Research Online Documents on Economics 6635, London School of Economics and Political Science, LSE Library.
    7. Axel Munk & Nicolai Bissantz & Thorsten Wagner & Gudrun Freitag, 2005. "On difference‐based variance estimation in nonparametric regression when the covariate is high dimensional," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 19-41, February.
    8. Spokoiny, Vladimir, 2002. "Variance Estimation for High-Dimensional Regression Models," Journal of Multivariate Analysis, Elsevier, vol. 82(1), pages 111-133, July.
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    Cited by:

    1. Paola Gloria Ferrario, 2018. "Partitioning estimation of local variance based on nearest neighbors under censoring," Statistical Papers, Springer, vol. 59(2), pages 423-447, June.

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