1 Nonparametric Regression Given data of the form(1,y),(2,y2),...,(n,Un),we seek an estimate of the regression function g(r)satisfying the model y=g(x)+8 where the noise term satisfies the usual conditions assumed for simple linear regression. There are several approaches to this problem.In this chapter,we will describe methods involving splines as well as local polynomial methods. 1.1 Spline Regression The discovery that piecewise polynomials or splines could be used in place of polynomials occurred in the early twentieth century.Splines have since become one of the most popular ways of approximating nonlinear functions. In this section,we will describe some of the basic properties of splines,describing two bases.We will then go on to discuss how to estimate coefficients of a spline using least-squares regression.We close this section with a discussion of smoothing splines. 1.1.1 Basic properties of splines Splines are essentially defined as piecewise polynomials.In this subsection,we will de- scribe how splines can be viewed as linear combinations of truncated power functions. We will then describe the B-spline basis which is a more convenient basis for computing splines. Truncated power functions Let t be any real number,then we can define a pth degree truncated power function as (x-t)=(z-t)PI>t)() As a function of x,this function takes on the value 0 to the left of t,and it takes on the value (x-t)P to the right of t.The number t is called a knot. The above truncated power function is a basic example of a spline.In fact,it is a member of the set of basis functions for the space of splines. 11 Nonparametric Regression Given data of the form (x1, y1),(x2, y2), . . . ,(xn, yn), we seek an estimate of the regression function g(x) satisfying the model y = g(x) + ε where the noise term satisfies the usual conditions assumed for simple linear regression. There are several approaches to this problem. In this chapter, we will describe methods involving splines as well as local polynomial methods. 1.1 Spline Regression The discovery that piecewise polynomials or splines could be used in place of polynomials occurred in the early twentieth century. Splines have since become one of the most popular ways of approximating nonlinear functions. In this section, we will describe some of the basic properties of splines, describing two bases. We will then go on to discuss how to estimate coefficients of a spline using least-squares regression. We close this section with a discussion of smoothing splines. 1.1.1 Basic properties of splines Splines are essentially defined as piecewise polynomials. In this subsection, we will de￾scribe how splines can be viewed as linear combinations of truncated power functions. We will then describe the B-spline basis which is a more convenient basis for computing splines. Truncated power functions Let t be any real number, then we can define a pth degree truncated power function as (x − t) p + = (x − t) p I{x>t}(x) As a function of x, this function takes on the value 0 to the left of t, and it takes on the value (x − t) p to the right of t. The number t is called a knot. The above truncated power function is a basic example of a spline. In fact, it is a member of the set of basis functions for the space of splines. 1