Kernel regression when the boundary region is large, with an application to testing the adequacy of polynomial models.

*(English)*Zbl 0764.62036Summary: It is well known that kernel regression estimators are subject to so- called boundary or edge effects, a phenomenon in which the bias of an estimator increases near the endpoints of the estimation interval. When the regression curve is linear or nearly linear, the requisite amount of smoothing is so great that the boundary region is effectively the entire estimation interval. Special boundary kernels are proposed here to deal with such cases.

It is shown that the proposed kernel estimator has a property also enjoyed by cubic smoothing splines; namely, as the estimator’s smoothing parameter becomes large, the estimator tends to a straight line. The limiting straight line is essentially the least squares line when the design points are equally spaced. A simple generalization of ideas in the linear case leads to kernel estimates that are polynomials of any given degree for large bandwidths. Such estimates are an important component of a proposed test for the adequacy of a polynomial model. The test statistic is the bandwidth chosen to minimize an estimated risk function. An example illustrates the usefulness of the new boundary kernels.

It is shown that the proposed kernel estimator has a property also enjoyed by cubic smoothing splines; namely, as the estimator’s smoothing parameter becomes large, the estimator tends to a straight line. The limiting straight line is essentially the least squares line when the design points are equally spaced. A simple generalization of ideas in the linear case leads to kernel estimates that are polynomials of any given degree for large bandwidths. Such estimates are an important component of a proposed test for the adequacy of a polynomial model. The test statistic is the bandwidth chosen to minimize an estimated risk function. An example illustrates the usefulness of the new boundary kernels.