Estimation and inference for error variance in bivariate nonparametric regression

In nonparametric regression, principal interest usually lies in the estimation of a smooth curve or surface defining the mean response at particular values of the co-variates. However, in order to move beyond descriptive use an estimate of the underlying error variance is required as an essential step in constructing interval estimates or comparing models. In the case of a single covariate, a variety of methods is available for estimating error variance, based on local differencing techniques. These methods are investigated in the important case of two covariates. An estimator constructed from residuals based on a nonparametric regression surface using a very small value of smoothing parameter is proposed as the most effective approach. A specific proposal based on nearest neighbour distances is investigated for choice of the smoothing parameter. Quadratic form techniques are used to construct confidence intervals and tests based on estimates of error variance. These include comparisons of error variances from different groups of data and the assessment of the assumption of constant variance across a regression surface. The techniques are illustrated on examples with real data.