Local Regression Models

Local regression models are regression models where the parameters are ‘localized’, that is, they are allowed to vary with some or all of the covariates in a general way. Suppose that (Y, X) are random variables and let 1<math display='block'> <mrow> <mi>E</mi><mrow><mo>(</mo> <mrow> <mi>Y</mi><mo>&#x007C;</mo><mi>X</mi><mo>=</mo><mi>x</mi> </mrow> <mo>)</mo></mrow><mo>=</mo><mi>m</mi><mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow> </mrow> </math>$$E\left( {Y|X=x} \right)=m\left( x \right)$$ when it exists. The regression function m(x) is of primary interest because it describes how X affects Y One may also be interested in derivatives of m or averages thereof or in derived quantities like conditional variance var(Y|X = x) = E(2|X = x) − E2 (Y\X = x). In cases of heavy-tailed distributions, the conditional expectation may not exist, in which case one may instead work with other location functionals like trimmed mean or median. The conditional expectation is particularly easy to deal with but a lot of what is done for the mean can also be done for the median or other quantities.