The limit of the partial sums process of spatial least squares residuals

Abstract

AMS subject classifications:Keywords: Spatial least squares residuals Partial sums process Set indexed partial sums Brownian sheet Brownian pillow Bounded variation Riemann-Stieltjes integral a b s t r a c t We establish a functional central limit theorem for a sequence of least squares residuals of spatial data from a linear regression model. Under mild assumptions on the model we explicitly determine the limit process in the case where the assumed linear model is true. Moreover, in the case where the assumed linear model is not true we explicitly establish the limit process for the localized true regression function under mild conditions. These results can be used to develop non-parametric model checks for linear regression. Our proofs generalize ideas of a univariate geometrical approach due to Bischoff [W. Bischoff, The structure of residual partial sums limit processes of linear regression models, Theory Stoch. Process. 8 (24) (2002) 23-28] which is different to that proposed by MacNeill and Jandhyala [I.B. MacNeill, V.K. Jandhyala, Change-point methods for spatial data, in: G.P. Patil, et al. (Eds.), Multivariate Environmental Statistics. Papers Presented at the 7th International Conference on Multivariate Analysis held at 33-53] established the limit process of set indexed partial sums of regression residuals. In our framework we get that result as an immediate consequence of a result of Alexander and Pyke [K.S. Alexander, R. Pyke, A uniform central limit theorem for set-indexed partial-sum processes with finite variance, Ann. Probab. 14 (1986) 582-597]. The reason for that is that by our geometrical approach we recognize the structure of the limit process: it is a projection of the Brownian sheet onto a certain subspace of the reproducing kernel Hilbert space of the Brownian sheet. Several examples are discussed.

On Weak Convergence of the Partial Sums Processes of the Least Squares Residuals of Multivariate Spatial Regression

“…Arnold [2], and Christensen [10], p. [8][9], where the last equation follows from the definition of the component wise projection. Here and throughout the paper P V and P V ⊥ = Id − P V denote the orthogonal projectors onto a linear subspace V and onto the orthogonal complement of V, respectively.…”

“…[see also Park [17] and [9] for the definition of T n ]. The space C p (I) and (R n×n ) p are the product of the p copies of C(I) and R n×n , respectively, where C p (I) is endowed with a topology induced by a metric ρ, defined by…”

“…Therefore in the classical theory of model check for multivariate linear regression model the question whether the assumed model holds true is tasted based on the length of the residuals (cf. Arnold [2], and Christensen [10], p. [9][10][11][12][13][14][15][16][17][18][19][20]. Since the partial sums operator is oneto-one on (R n×n ) p , instead of investigating the length of the residuals we observe the length of the partial sums of the residuals such as CM n .…”

“…The equality T n (P Wn B n×n ) = P WnH B T n (B n×n ), for every B n×n ∈ R n×n , was investigated in Proposition 2.2 of [9], where W nHB := {T n (A n×n ) : A n×n ∈ W n }. Hence by using the definition of T n×n×p and the component wise projection, it holds…”

“…The correctness of the assumed models and the existence of a change in the regression function or in the parameters of the model were detected by means of the Kolmogorov, Kolmogorov-Smirnov, and the Cramé-von Mises functionals of the PSPR. In MacNeill [16] and Xie [22], the application of the PSPR has been extended to the problem of boundary detection for SLRM, whereas in Bischoff and Somayasa [9], and Somayasa [18] and [19], it has been investigated from the perspective of model check for the SLRM.…”