ABSTRACT
Although the main work in nonparametric regression considers the problem of estimating the regression curve g itself, methods for estimating the variance have been discussed by numerous authors in recent years. To our knowledge, Breiman and Meisel (1976) were the first who considered the problem of estimating the variance in nonparametric regression as a topic in its own right. Their motivation was twofold. On the one hand, if the varia bility of the data is known, this can be used to judge the goodness-of-fit of a specified regression function. On the otherhand, such knowledge is useful for selecting a subset of the independent variables which best determine the dependent variable in a setup with high dimensional design space T. The variance is also required for the plug-in estimation of the bandwidth in nonparametric regression (see, e.g., Rice, 1984; Muller and Stadtmiiller, 1987a) and as a measure for the accuracy of a prespecified linear regression model to be validated (see, e.g., Hardle and Marron, 1990; Eubank and Hart, 1992; Dette and Munk, 1998). For further applications of variance estimators such as in quality control, in immunoassay, or in calibration of variance estimation see Carroll and Ruppert (1988).
