Smoothing Splines

In this section, we begin the study of nonparametric regression by way of smoothing splines. We wish to estimate the regression function f _o on a bounded interval, which we take to be [0, 1], from the data y _{1, n} , …, y _{n, n} , following the model Here, the x _in are design points (in this chapter, the design is deterministic) and d _n = (d _{1, n} , d _{2, n} , …, d _{n, n} )^T is the random noise. Typical assumptions are that d _{1, n} , d _{2, n} , …, d _{n, n} are uncorrelated random variables, with mean 0 and common variance, i.e., where σ is typically unknown. We refer to this as the Gauss-Markov model, in view of the Gauss-Markov theorem for linear regression models. At times, we need the added condition that for some k > 2. A typical choice is k = 4.