A framework to understand the asymptotic properties of Kriging and splines Eva M. Furrer and Douglas W. Nychka Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307 Abstract: Kriging is a nonparametric regression method used in geostatistics for estimating curves and surfaces for spatial data. It may come as a surprise that the Kriging estimator, normally derived as the best linear unbiased estimator, is also the solution of a particular variational problem. Thus, Kriging estimators can also be interpreted as generalized smoothing splines where the roughness penalty is determined by the covariance function of a spatial process. We build off the early work by Silverman (1982, 1984) and the analysis by Cox (1983, 1984), Messer (1991), Messer and Goldstein (1993) and others and develop an equivalent kernel interpretation of geostatistical estimators. Given this connection we show how a given covariance function influences the bias and variance of the Kriging estimate as well as the mean squared prediction error. Some specific asymptotic results are given for one dimensional corresponding Mat´ ern covariances that have as their limit cubic smoothing splines. Key words: Kriging, spline, equivalent kernel, asymptotic mean squared error 1 Introduction A common method in the analysis of spatial data is a geostatistical estimator known as Kriging. Although Kriging is typically derived as a best linear unbiased estimator it can also be viewed as a nonparametric curve and surface estimator. Given this later perspective, it is of interest to understand Kriging in terms of the large sample properties such as the asymptotic variance and bias that are well established for kernel estimators. The key idea developed in this work is that Kriging estimators can be interpreted as generalized splines and the asymptotic techniques similar to those described in Nychka (1995) can be brought to bear on the Kriging estimators. The problem one is faced with in nonparametric regression is to estimate an unknown function, g, on [0, 1], for which the observations y i are supposed to depend on the “locations” x i following the model: y i = g(x i )+ ε i ,i =1,...,n, where the ε i are iid random errors with common variance σ 2 . Note that we use the interval [0, 1] without loss of generality. The solution to this problem by either spline or kernel methods can be written as ˆ g(x)= 1 n n  i=1 ω(x,x i )y i , (1) 1