The Annals of Statistics

Fractals with point impact in functional linear regression

Ian W. McKeague and Bodhisattva Sen

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This paper develops a point impact linear regression model in which the trajectory of a continuous stochastic process, when evaluated at a sensitive time point, is associated with a scalar response. The proposed model complements and is more interpretable than the functional linear regression approach that has become popular in recent years. The trajectories are assumed to have fractal (self-similar) properties in common with a fractional Brownian motion with an unknown Hurst exponent. Bootstrap confidence intervals based on the least-squares estimator of the sensitive time point are developed. Misspecification of the point impact model by a functional linear model is also investigated. Non-Gaussian limit distributions and rates of convergence determined by the Hurst exponent play an important role.

Article information

Ann. Statist., Volume 38, Number 4 (2010), 2559-2586.

First available in Project Euclid: 11 July 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62E20: Asymptotic distribution theory 62M09: Non-Markovian processes: estimation
Secondary: 60J65: Brownian motion [See also 58J65]

Functional linear regression fractional Brownian motion M-estimation misspecification nonstandard asymptotics empirical processes bootstrap methods


McKeague, Ian W.; Sen, Bodhisattva. Fractals with point impact in functional linear regression. Ann. Statist. 38 (2010), no. 4, 2559--2586. doi:10.1214/10-AOS791.

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