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Functional central limit theorems for self-normalized least squares processes in regression with possibly infinite variance data

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  • Csörgő, Miklós
  • Martsynyuk, Yuliya V.
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    Abstract

    Based on an R2-valued random sample {(yi,xi),1≤i≤n} on the simple linear regression model yi=xiβ+α+εi with unknown error variables εi, least squares processes (LSPs) are introduced in D[0,1] for the unknown slope β and intercept α, as well as for the unknown β when α=0. These LSPs contain, in both cases, the classical least squares estimators (LSEs) for these parameters. It is assumed throughout that {(x,ε),(xi,εi),i≥1} are i.i.d. random vectors with independent components x and ε that both belong to the domain of attraction of the normal law, possibly both with infinite variances. Functional central limit theorems (FCLTs) are established for self-normalized type versions of the vector of the introduced LSPs for (β,α), as well as for their various marginal counterparts for each of the LSPs alone, respectively via uniform Euclidean norm and sup–norm approximations in probability. As consequences of the obtained FCLTs, joint and marginal central limit theorems (CLTs) are also discussed for Studentized and self-normalized type LSEs for the slope and intercept. Our FCLTs and CLTs provide a source for completely data-based asymptotic confidence intervals for β and α.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 121 (2011)
    Issue (Month): 12 ()
    Pages: 2925-2953

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    Handle: RePEc:eee:spapps:v:121:y:2011:i:12:p:2925-2953

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    Related research

    Keywords: Simple linear regression; Domain of attraction of the normal law; Infinite variance; Slowly varying function at infinity; Studentized/self-normalized least squares estimator/process; Cholesky square root of a matrix; Symmetric positive definite square root of a matrix; Standard/bivariate Wiener process; Functional central limit theorem; Sup–norm approximation in probability; Direct product of two measurable spaces; Uniform Euclidean norm approximation in probability; Asymptotic confidence interval; Signal-to-noise ratio; Generalized domain of attraction of the d-variate normal law;

    References

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    1. Rackauskas, Alfredas & Suquet, Charles, 2001. "Invariance principles for adaptive self-normalized partial sums processes," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 63-81, September.
    2. Vu, H. T. V. & Maller, R. A. & Klass, M. J., 1996. "On the Studentisation of Random Vectors," Journal of Multivariate Analysis, Elsevier, vol. 57(1), pages 142-155, April.
    3. Maller, R. A., 1993. "Quadratic Negligibility and the Asymptotic Normality of Operator Normed Sums," Journal of Multivariate Analysis, Elsevier, vol. 44(2), pages 191-219, February.
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    Cited by:
    1. Martsynyuk, Yuliya V., 2012. "Invariance principles for a multivariate Student process in the generalized domain of attraction of the multivariate normal law," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2270-2277.

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