Functional linear regression with points of impact
| Autor: | Dominik Poß, Pascal Sarda, Alois Kneip |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Statistics and Probability
model selection Generalization Mathematics - Statistics Theory 02 engineering and technology Statistics Theory (math.ST) 01 natural sciences Functional linear regression 010104 statistics & probability 62G08 62J05 Linear regression 0202 electrical engineering electronic engineering information engineering FOS: Mathematics 0101 mathematics Brownian motion Variable (mathematics) Mathematics Fractional Brownian motion Stochastic process Mathematical analysis Scalar (physics) 020206 networking & telecommunications 62M99 Identifiability stochastic processes Statistics Probability and Uncertainty nonstandard asymptotics |
| Zdroj: | Ann. Statist. 44, no. 1 (2016), 1-30 |
| DOI: | 10.48550/arxiv.1601.02798 |
| Popis: | The paper considers functional linear regression, where scalar responses $Y_1,\ldots,Y_n$ are modeled in dependence of i.i.d. random functions $X_1,\ldots,X_n$. We study a generalization of the classical functional linear regression model. It is assumed that there exists an unknown number of "points of impact," that is, discrete observation times where the corresponding functional values possess significant influences on the response variable. In addition to estimating a functional slope parameter, the problem then is to determine the number and locations of points of impact as well as corresponding regression coefficients. Identifiability of the generalized model is considered in detail. It is shown that points of impact are identifiable if the underlying process generating $X_1,\ldots,X_n$ possesses "specific local variation." Examples are well-known processes like the Brownian motion, fractional Brownian motion or the Ornstein-Uhlenbeck process. The paper then proposes an easily implementable method for estimating the number and locations of points of impact. It is shown that this number can be estimated consistently. Furthermore, rates of convergence for location estimates, regression coefficients and the slope parameter are derived. Finally, some simulation results as well as a real data application are presented. Comment: Published at http://dx.doi.org/10.1214/15-AOS1323 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org) |
| Databáze: | OpenAIRE |
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