Content deleted Content added
Line 36:
Assuming that <math>\mathcal{S} = \mathcal{T}</math>, another model, known as the functional concurrent model, sometimes also referred to as the varying-coefficient model, is of the form
{{NumBlk|::|<math display="block">Y(t) = \alpha_0(t) + \alpha(t)X(t)+\varepsilon(t),\ \text{for}\ t\in\mathcal{T},</math>|{{EquationRef|6}}}}
where <math>\alpha_0</math> and <math>\alpha</math> are coefficient functions. Note that model ({{EquationNote|6}}) assumes the value of <math>Y</math> at time <math>t</math>, i.e., <math>Y(t)</math>, only depends on that of <math>X</math> at the same time, i.e., <math>X(t)</math>. Various estimation methods can be applied to model ({{EquationNote|6}}).<ref>{{Cite journal |last=Fan |first=Jianqing |last2=Zhang |first2=Wenyang |date= |title=Statistical estimation in varying coefficient models |url=https://projecteuclid.org/journals/annals-of-statistics/volume-27/issue-5/Statistical-estimation-in-varying-coefficient-models/10.1214/aos/1017939139.full |journal=The Annals of Statistics |volume=27 |issue=5 |pages=1491–1518 |doi=10.1214/aos/1017939139 |issn=0090-5364}}</ref><ref>{{Cite journal |last=Huang
Adding multiple functional covariates, model ({{EquationNote|6}}) can also be extended to
<math display="block">Y(t) = \alpha_0(t) + \sum_{j=1}^p\alpha_j(t)X_j(t)+\varepsilon(t),\ \text{for}\ t\in\mathcal{T},</math>
| |||