On the use of the variogram in checking for independence in spatial data

The variogram is a standard tool in the analysis of spatial data, and its shape provides useful information on the form of spatial correlation that may be present. However, it is also useful to be able to assess the evidence for the presence of any spatial correlation. A method of doing this, based on an assessment of whether the true function underlying the variogram is constant, is proposed. Nonparametric smoothing of the squared differences of the observed variables, on a suitably transformed scale, is used to estimate variogram shape. A statistic based on a ratio of quadratic forms is proposed and the test is constructed by investigating the distributional properties of this statistic under the assumption of an independent Gaussian process. The power of the test is investigated. Reference bands are proposed as a graphical follow-up. An example is discussed.     

On partial local smoothing rules for curve estimation

We compare the performances of local and global rules for smoothingparameter choice, in terms of asymptotic mean squared errorsof the resulting estimators. In some instances there is surprisinglylittle to choose between local and global approaches; our analysisidentifies contexts where the differences are small or large.This work motivates development of smoothing rules that forma ‘half-way house’ between local and global smoothing.There, interpolation provides a basis for partial local smoothing.A key result shows that interpolation on even a coarse gridcan produce a very good approximation to full local smoothing.Our theoretical and numerical results lead us to suggest linearinterpolation of a bandwidth obtained by integral approximationson discrete intervals.     

Reducing variability of crossvalidation for smoothing-parameter choice

One of the attractions of crossvalidation, as a tool for smoothing-parameterchoice, is its applicability to a wide variety of estimatortypes and contexts. However, its detractors comment adverselyon the relatively high variance of crossvalidatory smoothingparameters, noting that this compromises the performance ofthe estimators in which those parameters are used. We show thatthe variability can be reduced simply, significantly and reliablyby employing bootstrap aggregation or bagging. We establishthat in theory, when bagging is implemented using an adaptivelychosen resample size, the variability of crossvalidation canbe reduced by an order of magnitude. However, it is arguablymore attractive to use a simpler approach, based for exampleon half-sample bagging, which can reduce variability by approximately50%.