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A functional multiple index model is given by
<math display="block">Y = g\left(\int_{\mathcal{T}} X^c(t) \beta_1(t)dt, \cdots, \int_{\mathcal{T}} X^c(t) \beta_p(t)dt \right) + \epsilon.</math>
Taking <math>p=1</math> yields a functional single index model. However, this model is problematic due to [[Curse of dimensionality|curse of dimensionality]]. With <math>p>1</math> and relatively small sample sizes, the estimator given by this model often has large variance<ref name=chen:11>Chen, Hall and Müller (2011). "Single and multiple index functional regression models with nonparametric link". ''The Annals of Statistics''. '''39''' (3):1720–1747. [[Digital object identifier|doi]]:[http://doi.org/10.1214/11-AOS882 10.1214/11-AOS882].</ref>. An alternative <math>p</math>-component functional multiple index model can be expressed as
<math display="block">Y = g_1\left(\int_{\mathcal{T}} X^c(t) \beta_1(t)dt\right)+ \cdots+ g_p\left(\int_{\mathcal{T}} X^c(t) \beta_p(t)dt \right) + \epsilon.</math>
Estimation methods for functional single and multiple index models are available<ref name=chen:11/><ref>Jiang and Wang (2011). "Functional single index models for longitudinal data". '''39''' (1):362–388. [[Digital object identifier|doi]]:[http://doi.org/10.1214/10-AOS845 10.1214/10-AOS845].</ref>.
=== Functional additive models (FAMs) ===
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