Local regression: Difference between revisions

Revision as of 03:59, 18 February 2025

Revision as of 04:04, 18 February 2025

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Extensive theoretical work continued to appear throughout the 1990's. Important contributions include [[Jianqing Fan]] and [[Irène Gijbels]] (1992)<ref>{{citeQ|Q132202273}}</ref> studying efficiency properties, and [[David Ruppert]] and [[Matthew P. Wand]] (1994)<ref>{{citeQ|Q132202598}}</ref> developing an asymptotic distribution theory for multivariate local regression.

An important extension of local regression is Local Likelihood Estimation, formulated by [[Robert Tibshirani]] and [[Trevor Hastie]] (1987).<ref name="tib-hast-lle">{{citeQ|Q132187702}}</ref> This replaces the local least-squares criterion with a likelihood-based criterion, thereby extending the local regresion method to the [[Generalized linear model]] setting; for example binary data; count data; censored data.

Practical implementations of local regression began appearing in statistical software in the 1980's. Cleveland (1981)<ref>{{citeQ|Q29541549}}</ref> introduces the LOWESS routines, intended for smoothing scatterplots. This implements local linear fitting with a single predictor variable, and also introduces robustness downweighting to make the procedure resistant to outliers. An entirely new implementation, LOESS, is described in Cleveland and [[Susan J. Devlin]] (1988)<ref name="clevedev">{{citeQ|Q29393395}}</ref>. LOESS is a multivariate smoother, able to handle spatial data with two (or more) predictor variables, and uses (by default) local quadratic fitting. Both LOWESS and LOESS are implemented in the [[S (programming language)|S]] and [[R (programming language)|R]] programming languages. See also Cleveland's Local Fitting Software.<ref>{{cite web |last=Cleveland|first=William|title=Local Fitting Software|url=https://web.archive.org/web/20050912090738/http://www.stat.purdue.edu/~wsc/localfitsoft.html}}</ref>

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====Local Likelihood Estimation====

In local likelihood estimation, developed in HastieTibshirani and TibshiraniHastie (1987)<ref name=htloclike"tib-hast-lle" />, the observations <math>Y_i</math> are assumed to come from a parametric family of distributions, with a known probability density function (or mass function, for discrete data),

<math display="block">

Y_i \sim f(y,\theta(x_i)),