STA305H1 Lecture Notes - Lecture 8: Local Regression, Nonparametric Regression, Nonparametric Statistics
18 Jan 2018
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Suppose that we observe (x1, y1), , (xn, yn) where {xi} are the values of some predictor variable and {yi} are the values of some response variable with the two variables related via. Yi = g(xi) + for i = 1, , n; we assume that { i} are (unobserved) random variables with mean 0 and nite variance. In non-parametric regression, we make weaker assumptions about the function g; for example, rather than assuming a particular form for g (such as a polynomial), we may assume that g is a smooth (that is, di erentiable) function. S(x) = {i : |xi x| h} and then our estimate of g(x) is where the weights {wi(x)} satisfy wi(x)yi (cid:2)g(x) = (cid:3)i s(x) (cid:3)i s(x) wi(x) = 1. As stated in lecture, the bandwidth h (for a given method) controls the smoothness of the estimated function(cid:2)g(x) with the smoothness increasing with the bandwidth.
