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    Nonparametric regression: a brief overview and recent developments
    (Maltepe Üniversitesi, 2009) Swanepoel, C. J.
    A regression curve describes a general relationship between two or more quantitative variables. In a multivariate situation vectors of explanatory variables as well as response variables may be present. For the simple case of one-dimensional explanatory and response variables, n data points S := {(Xi, Yi), i = 1, 2, . . . , n} are collected. The regression relationship can be modeled by Yi = m(Xi) + ?i, i = 1, 2, . . . , n, where m(x) = E(Y |X = x) is the unknown regression function and the ?i’s are independent random errors with mean 0 and unknown variance ? 2 . Nonparametric methods relax on traditional assumptions and usually only assumes that m belongs to an infinite-dimensional collection of smooth functions. Several popular nonparametric estimators are discussed, mostly of the form ˆm(x) = 1 n Xn i=1 Wn,i(x)Yi, where {Wn,i} n i=1 denotes a sequence of weights depending on the explanatory variables. Several kernel and nearest neighbour approaches to the weight functions are considered. Each of these estimators depends on a smoothing parameter and the issue of estimating it is discussed briefly. The performance of ˆm(x) is assessed via methods involving the mean squared error (MSE) and the mean integrated squared error (M ISE). Two recent developments of improving the performance of ˆm(x) are discussed, namely “boosting” and ‘bagging”, which are respectively an iterative computer intensive method, and an averaging method involving the generation of bootstrap samples. These methods, together with variations of these methods, for example the method referred to as “bragging”, are illustrated.