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Fractals with point impact in functional linear regression. (English) Zbl 1196.62116
Summary: 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.

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G08 Nonparametric regression and quantile regression
62M09 Non-Markovian processes: estimation
62E20 Asymptotic distribution theory in statistics
62J05 Linear regression; mixed models
60J65 Brownian motion
Software:
longmemo; fda (R)
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