Fhatx

Fhatx is a notation used in statistics, data analysis, and signal processing to denote the estimator of a function f evaluated at a point x. In plain text, the form fhatx (often written as f̂(x) in typeset material) represents the value predicted by a model for the unknown function f at x, given observed data. Because the estimator depends on the data and the chosen method, fhatx is not a single object but a family of estimators that approximate f.

Common constructions of fhatx include both nonparametric and parametric regression. In nonparametric regression, kernel methods yield fhatx as the Nadaraya-Watson estimator, for example: fhat(x) = sum_i K((x − X_i)/h) Y_i / sum_i K((x − X_i)/h). Parametric methods provide fhatx as the fitted value from models such as linear, polynomial, or generalized additive models. In time series and signal processing, fhatx may denote the reconstructed or denoised value of the signal at time x.

Key properties of fhatx concern how well it approximates the true function f. These include bias, variance, consistency, and the overall mean squared error, which depend on the method and data, as well as tuning parameters like bandwidth, degree, or regularization strength. Assessing performance typically involves cross-validation, held-out data, or information criteria.

See also: estimator, regression, kernel smoothing, smoothing splines, f-hat notation. Further reading would include standard texts on nonparametric regression and statistical estimation.