Each of the smoothing methods discussed in the previous section has one or more `smoothing parameters' that control the amount of smoothing being performed. For example, the bandwidth in the kernel smoother or local regression methods, and the parameter in the penalized likelihood criterion. In implementing the smoothers, the first question to be asked is how should the smoothing parameters be chosen? More generally, how can the performance of a smoother with given smoothing parameters be assessed? A deeper question is in comparing fits from different smoothers. For example, we have seen for the fuel economy dataset that a local linear fit with (Fig. 5.2) produces a fit similar to a smoothing spline with (Fig. 5.3). Somehow, we want to be able to say these two smoothing parameters are equivalent.
As a prelude to studying methods for bandwidth selection and other statistical inference procedures, we must first study some of the properties of linear smoothers. We can consider measures of goodness-of-fit, such as the mean squared error,
Intuitively, as the bandwidth increases, more data is used to construct the estimate , and so the variance decreases. On the other hand, the local polynomial approximation is best over small intervals, so we expect the bias to increase as the bandwidth increases. Choosing is a tradeoff between small bias and small variance, but we need more precise characterizations to derive and study selection procedures.
The bias of a linear smoother is given by
For illustration, consider the bias of the local linear regression estimate defined by (5.6). A three-term Taylor series gives
The next step is to approximate summations by integrals, both in (5.12) and in the matrix equation (5.9) defining . This leads to
Bias expansions like (5.13) are derived much more generally by ; their results cover arbitrary degree local polynomials and multidimensional fits also. Their results imply that when , the degree of the local polynomial, is odd, the dominant term of the bias is proportional to . When is even, the first-order term can disappear, leading to bias of order .
To derive the variance of a linear smoother, we need to make assumptions about the random errors in (5.1). The most common assumption is that the errors are independent and identically distributed, with variance . The variance of a linear smoother (5.3) is
As with bias, informative approximations to the variance can be derived by replacing sums by integrals. For local linear regression, this leads to
Under the model (5.1) the observation has variance , while the estimate has variance . The quantity measures the variance reduction of the smoother at a data point . At one extreme, if the `smoother' interpolates the data, then and . At the other extreme, if , . Under mild conditions on the weight function, a local polynomial smoother satisfies
A global measure of the amount of smoothing is provided by
An alternative representation of is as follows. Let be the `hat matrix', which maps the data to fitted values:
The diagonal elements of , provide another measure of the amount of smoothing at . If the smooth interpolates the data, then is the corresponding unit vector with . If the smooth is simply the global average, . The corresponding definition of degrees of freedom is
For the local linear regression in Fig. 5.2, the degrees of freedom are and . For the smoothing spline smoother in Fig. 5.3, and . By either measure the degrees of freedom are similar for the two fits. The degrees of freedom provides a mechanism by which different smoothers, with different smoothing parameters, can be compared: we simply choose smoothing parameters producing the same number of degrees of freedom. More extensive discussion of the degrees of freedom of a smoother can be found in  and .
The final component needed for many statistical procedures is an estimate of the error variance . One such estimate is