Asymptotic Theory in Model Diagnostic for General Multivariate Spatial Regression

Abstract

We establish an asymptotic approach for checking the appropriateness of an assumed multivariate spatial regression model by considering the set-indexed partial sums process of the least squares residuals of the vector of observations. In this work, we assume that the components of the observation, whose mean is generated by a certain basis, are correlated. By this reason we need more effort in deriving the results. To get the limit process we apply the multivariate analog of the well-known Prohorov's theorem. To test the hypothesis we define tests which are given by Kolmogorov-Smirnov (KS) and Cramér-von Mises (CvM) functionals of the partial sums processes. The calibration of the probability distribution of the tests is conducted by proposing bootstrap resampling technique based on the residuals. We studied the finite sample size performance of the KS and CvM tests by simulation. The application of the proposed test procedure to real data is also discussed.

Accessing the Appropriateness of a Spatial Regression Using Generalized (<i>h</i>₁<i>h</i>₂-)Slepian Random Field

“…This approach will be useful for handling the modeling problem of spatial data. In a forthcoming paper of Somayasa and et al the investigation is extended to multivariate spatial regression model defined in (Somayasa et al, 2015b;Somayasa and Wibawa, 2015;Somayasa et al, 2016) by considering the moving sum process of the multivariate recursive residuals.…”

“…Cressie (1993;Ripley, 2004;Wackernagel, 2003). The measured variable in spatial analysis might stand for percentage of either Ni, Fe or Au in mining exploration, see e.g., Tahir (2010;Somayasa et al, 2015a;2015b;Somayasa and Wibawa, 2015;Somayasa et al, 2016) or the incidence rates for breast cancer, cf. MacNeill et al (1994).…”

Accessing the Power of Tests Based on Set-Indexed Partial Sums of Multivariate Regression Residuals

“…Investigating the partial sums of least squares residuals has been shown to be reasonable and powerful tool for testing the adequacy of an assumed multivariate regression model; see Somayasa and et al [1][2][3][4]. The development of the technique was motivated by the works proposed mainly for the purpose of detecting change in parameter as well as for detecting the existence of boundaries in univariate spatial regression; see [5][6][7][8] for references.…”

“…, where for any ∈ 2 ( 0 , G), is defined as ( ) fl ∫ 0 . Under mild condition, [1][2][3] showed after a suitable localization given to the regression function that the sequence of the partial sums of the least squares residuals obtained from the multivariate regression model Y (t) = g (t) + E (t) , t fl ( 1 , . .…”

“…Throughout the paper Σ −1/2 W ⊥ H Z g will be denoted by g and * W ⊥ H Z Z by W f, 0 , for the sake of brevity. It was established in [1][2][3] that W f, 0 is a projection of Z onto the orthogonal complement of W H Z which is a finite dimensional subspace of the so-called reproducing kernel Hilbert Space (RKHS) of Z , denoted by H Z , given by…”