Some linear models are necessarily parametric. (English) Zbl 0888.62098
Summary: We prove the surprising result that rather general assumptions on the set of admissible signals \(\xi(t)\) observed in the presence of additive noise \(\varepsilon(t)\) on a closed interval \([a,b]\) imply that the set is finite-dimensional, i.e., \(\xi(t)= \theta_1\xi_1(t)+\dots+ \theta_m\xi_m(t)\) for some integer \(m\geq 1\) and fixed functions \(\xi_1(t),\dots, \xi_m(t)\). Thus, estimating the signal \(\xi(t)\) from observations of \(x(t)= \xi(t)+ \varepsilon(t)\) reduces to estimating the parameters \(\theta_1,\dots, \theta_m\). This gives a strong argument in favor of parametric linear models.
MSC:
| 62M99 | Inference from stochastic processes |
| 62J05 | Linear regression; mixed models |
| 46N30 | Applications of functional analysis in probability theory and statistics |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
References:
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