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An important extension of local regression is Local Likelihood Estimation, formulated by [[Robert Tibshirani]] and [[Trevor Hastie]] (1987).<ref name="tib-hast-lle">{{cite Q|Q132187702}}</ref> This replaces the local least-squares criterion with a likelihood-based criterion, thereby extending the local regression 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 1980s. Cleveland (1981)<ref>{{cite Q|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">{{cite Q|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=http://www.stat.purdue.edu/~wsc/localfitsoft.html|archive-url=https://web.archive.org/web/20050912090738/http://www.stat.purdue.edu/~wsc/localfitsoft.html |archive-date=12 September 2005 }}</ref>
While Local Regression, LOWESS and LOESS are sometimes used interchangeably, this usage should be considered incorrect. Local Regression is a general term for the fitting procedure; LOWESS and LOESS are two distinct implementations.
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One question not addressed above is, how should the bandwidth depend upon the fitting point <math>x</math>? Often a constant bandwidth is used, while LOWESS and LOESS prefer a nearest-neighbor bandwidth, meaning ''h'' is smaller in regions with many data points. Formally, the smoothing parameter, <math>\alpha</math>, is the fraction of the total number ''n'' of data points that are used in each local fit. The subset of data used in each weighted least squares fit thus comprises the <math>n\alpha</math> points (rounded to the next largest integer) whose explanatory variables' values are closest to the point at which the response is being estimated.<ref name="NIST">NIST, [http://www.itl.nist.gov/div898/handbook/pmd/section1/pmd144.htm "LOESS (aka LOWESS)"], section 4.1.4.4, ''NIST/SEMATECH e-Handbook of Statistical Methods,'' (accessed 14 April 2017)</ref>
More sophisticated methods attempt to choose the bandwidth ''adaptively''; that is, choose a bandwidth at each fitting point <math>x</math> by applying criteria such as cross-validation locally within the smoothing window. An early example of this is [[Jerome H. Friedman]]'s<ref>{{citation|first=Jerome H.|last=Friedman|title=A Variable Span Smoother|date=October 1984|publisher=Technical report, Laboratory for Computational Statistics LCS 5; SLAC PUB-3466|doi=10.2171/1447470|doi-broken-date=20 March 2025 |url=http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-3477.pdf}}</ref> "supersmoother", which uses cross-validation to choose among local linear fits at different bandwidths.
===Degree of local polynomials===
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\sum_{i=1}^n w_i(x) \left | Y_i - \beta_0 - \ldots - \beta_p(x_i-x)^p \right |
</math>
results; this does not require a scale parameter. When <math>p=0</math>, this criterion is minimized by a locally weighted median; local <math>L_1</math> regression can be interpreted as estimating the ''median'', rather than ''mean'', response. If the loss function is skewed, this becomes local quantile regression. See [[Keming Yu]] and [[M. C. Jones (mathematician)|M.C. Jones]] (1998).<ref>{{citation |first1=Keming|last1=Yu|first2=M.C.|last2=Jones|title=Local Linear Quantile Regression|journal=Journal of the American Statistical Association|date=1998 |volume=93|issue=441 |pages=228–237|doi=10.1080/01621459.1998.10474104 }}</ref>
==Advantages==
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