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The limit of the partial sums process of spatial least squares residuals

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  • Bischoff, Wolfgang
  • Somayasa, Wayan

Abstract

We establish a functional central limit theorem for a sequence of least squares residuals of spatial data from a linear regression model. Under mild assumptions on the model we explicitly determine the limit process in the case where the assumed linear model is true. Moreover, in the case where the assumed linear model is not true we explicitly establish the limit process for the localized true regression function under mild conditions. These results can be used to develop non-parametric model checks for linear regression. Our proofs generalize ideas of a univariate geometrical approach due to Bischoff [W. Bischoff, The structure of residual partial sums limit processes of linear regression models, Theory Stoch. Process. 8 (24) (2002) 23-28] which is different to that proposed by MacNeill and Jandhyala [I.B. MacNeill, V.K. Jandhyala, Change-point methods for spatial data, in: G.P. Patil, et al. (Eds.), Multivariate Environmental Statistics. Papers Presented at the 7th International Conference on Multivariate Analysis held at Pennsylvania State University, University Park, PA, USA, May 5-9 1992, in: Ser. Stat. Probab., vol. 6, North-Holland, Amsterdam, 1993, pp. 289-306 (in English)]. Moreover, Xie and MacNeill [L. Xie, I.B. MacNeill, Spatial residual processes and boundary detection, South African Statist. J. 40 (1) (2006) 33-53] established the limit process of set indexed partial sums of regression residuals. In our framework we get that result as an immediate consequence of a result of Alexander and Pyke [K.S. Alexander, R. Pyke, A uniform central limit theorem for set-indexed partial-sum processes with finite variance, Ann. Probab. 14 (1986) 582-597]. The reason for that is that by our geometrical approach we recognize the structure of the limit process: it is a projection of the Brownian sheet onto a certain subspace of the reproducing kernel Hilbert space of the Brownian sheet. Several examples are discussed.

Suggested Citation

  • Bischoff, Wolfgang & Somayasa, Wayan, 2009. "The limit of the partial sums process of spatial least squares residuals," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2167-2177, November.
  • Handle: RePEc:eee:jmvana:v:100:y:2009:i:10:p:2167-2177
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    References listed on IDEAS

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    1. Wolfgang Bischoff & Enkelejd Hashorva & Jürg Hüsler & Frank Miller, 2003. "Exact asymptotics for Boundary crossings of the brownian bridge with trend with application to the Kolmogorov test," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(4), pages 849-864, December.
    2. anonymous, 1991. "Fed upgrades functional cost analysis program," Financial Update, Federal Reserve Bank of Atlanta, issue Win, pages 1-2,6.
    3. Wolfgang Bischoff & Enkelejd Hashorva & Jürg Hüsler & Frank Miller, 2005. "Analysis of a change-point regression problem in quality control by partial sums processes and Kolmogorov type tests," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 62(1), pages 85-98, September.
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    Cited by:

    1. Bischoff, W. & Gegg, A., 2011. "Partial sum process to check regression models with multiple correlated response: With an application for testing a change-point in profile data," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 281-291, February.
    2. Wayan Somayasa & Gusti N. Adhi Wibawa & La Hamimu & La Ode Ngkoimani, 2016. "Asymptotic Theory in Model Diagnostic for General Multivariate Spatial Regression," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2016, pages 1-16, September.
    3. Enkelejd Hashorva & Yuliya Mishura & Georgiy Shevchenko, 2021. "Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics," Journal of Theoretical Probability, Springer, vol. 34(2), pages 728-754, June.
    4. Pingjin Deng, 2016. "Asymptotic of Non-Crossings probability of Additive Wiener Fields," Papers 1610.07131, arXiv.org.

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