Testing Goodness-of-Fit of Parametric Spatial Trends ^†

Abstract

The aim of this work is to propose and analyze the behavior of a test statistic to assess a parametric trend surface, that is, a regression model with spatially correlated errors. The asymptotic behavior under the null hypothesis, as well as the asymptotic power of the test under local alternatives will be analyzed. Finite sample performance of the test is addressed by simulation, introducing a bootstrap calibration procedure.

1. Introduction

Consider a spatial stochastic process, which consists of a collection of random variables indexed on a certain domain of ${\mathbb{R}}^2$, with a well-defined joint distribution. In this framework, the observed data usually exhibit an important feature: close observations tend to be more similar than those which are far apart. Therefore, such observations cannot be treated as independent and the dependence structure should be taken into account in any descriptive or inferential procedure. In particular, from the perspective of spatial regression models (a trend surface plus an error term), the dependence structure should be considered and properly introduced into the model.

A common task in statistics is to determine whether a parametric model is an appropriate representation of a dataset. Under the assumption of independent errors, some authors have developed goodness-of-fit tests for parametric models that rely on a smooth alternative estimated by a nonparametric regression method, as [1] or [2].

A new proposal for testing a parametric trend surface is given in this paper. The proposed test is based on a comparison between a smooth version of a parametric fit with a nonparametric estimator of the trend (specifically, the multivariate local linear estimator will be used) in terms of a distance.

2. Statistical Model

Let $\left\{Z\right(\mathbf{s}),\mathbf{s}\in D\}$ be a random spatial process consisting of collections of random variables indexed in a domain $D\subset {\mathbb{R}}^2$ with a well-defined joint distribution. Consider n locations $\{{\mathbf{s}}_1,\cdots ,{\mathbf{s}}_n\}$ on the region D generated from a density f. The set of random variables corresponding with those locations will be represented by $\{Z\left({\mathbf{s}}_1\right),\cdots ,Z\left({\mathbf{s}}_n\right)\}$. Assume the model

$$Z\left({\mathbf{s}}_i\right)=m\left({\mathbf{s}}_i\right)+\epsilon \left({\mathbf{s}}_i\right),i=1,\cdots ,n,$$

(1)

where m is an unknown smooth regression function which is supposed to be twice continuously differentiable. The $\epsilon$ are unobserved random variables with

$$\mathbb{E}\left[\epsilon \left({\mathbf{s}}_i\right)\right]=0,\mathrm{Cov}(\epsilon \left({\mathbf{s}}_i\right),\epsilon \left({\mathbf{s}}_j\right))={\sigma }^2{\rho }_n({\mathbf{s}}_i-{\mathbf{s}}_j),i,j=1,\cdots ,n,$$

where ${\sigma }^2<\infty$ and ${\rho }_n$ is a continuous correlation function satisfying ${\rho }_n\left(0\right)=1$, ${\rho }_n\left(\mathbf{s}\right)={\rho }_n(-\mathbf{s})$ and $|{\rho }_n\left(\mathbf{s}\right)|\le 1$, $\forall \mathbf{s}$. The goal of this work is to test if the trend function belongs to a parametric family:

$$H_0:m\in {\mathcal{M}}_{\mathbf{\beta }}=\{m_{\mathbf{\beta }},\mathbf{\beta }\in \mathcal{B}\},\mathrm{vs}.H_a:m\notin {\mathcal{M}}_{\mathbf{\beta }},$$

(2)

with $\mathcal{B}\subset {\mathbb{R}}^p$ a compact set. One of the more usual approaches is to compare a smooth version of a parametric fit with a nonparametric estimator of $m\left(\mathbf{s}\right)$ and “thereafter” to reject $H_0$ if the distance between both fits exceeds a critical value.

3. Test Statistic

A suitable test statistic in order to solve the testing problem (2) could be computed as a weighted $L_2$—distance between the nonparametric and parametric fits, as in [2]:

$$T_n=n{\left|\mathbf{H}\right|}^{1/2}{\int }_D^{}{({\widehat{m}}_{\mathbf{H}}^{LL}\left(\mathbf{s}\right)-{\widehat{m}}_{\mathbf{H},\widehat{\mathbf{\beta }}}^{LL}\left(\mathbf{s}\right))}^{2}w\left(\mathbf{s}\right)d\mathbf{s},$$

(3)

where w is a weight function. A full definition of the elements of the test statistic $T_n$ can be found in Appendix A. For the calibration of the critical values, a bootstrap procedure is considered, see Appendix B.

4. Simulations

In this section, a simulation study showing the performance of the bootstrap procedure is presented. For this purpose, 500 samples of size $n=400$ are generated from an isotropic spatial process observed at regularly spaced locations $\{{\mathbf{s}}_1,\cdots ,{\mathbf{s}}_n\}$ in the unit square, where ${\mathbf{s}}_i=(s_{i1},s_{i2})$, $i=1,\cdots ,n$:

$$Z\left({\mathbf{s}}_i\right)=2+s_{i1}+s_{i2}+c{s}_{i1}^3+\epsilon \left({\mathbf{s}}_i\right),1\le i\le n.$$

(4)

The random errors $\epsilon \left({\mathbf{s}}_i\right)$ are normally distributed with zero mean and exponential covariance function $\mathrm{Cov}(\epsilon \left({\mathbf{s}}_i\right),\epsilon \left({\mathbf{s}}_j\right))={\sigma }^2\{exp(-\parallel {\mathbf{s}}_i-{\mathbf{s}}_j\parallel /a_e)\},$ with $\sigma =0.4$ and $\sigma =0.8$. Different values of parameter $a_e$ are considered: $a_e=0.4,0.6,0.8$. The bootstrap procedure has been performed using $B=500$ replicas for each sample. The weight function used was taken as $w\left(\mathbf{s}\right)=1$. For simplicity, the bandwidth matrix was considered $\mathbf{H}=\mathrm{diag}(h,h)$, and different bandwidth values were chosen, $h=0.10,0.15,0.20$.

In

Table 1. Proportion of rejections of the null hypothesis.

					h
$\mathit{\sigma }$	${\mathit{a}}_{\mathit{e}}$	c		0.10	0.15	$0.20$
$0.4$	0.4	0		0.052	0.047	$0.042$
		5		0.897	0.932	$0.911$
		10		0.905	0.948	$0.923$
$0.4$	0.6	0		0.054	0.042	$0.034$
		5		0.856	0.901	$0.898$
		10		0.894	0.926	$0.918$
$0.8$	0.8	0		0.068	0.048	$0.038$
		5		0.808	0.798	$0.806$
		10		0.845	0.803	$0.816$

, the simulated rejection probabilities obtained for $T_n$ are presented for the significance level $\alpha =0.05$ over the 500 trials. When c is equal to zero (under the null hypothesis of linearity of the trend), the proportion of rejections obtained is similar to the considered significance level, but this proportion depends directly on the value of the bandwidth h. When c is equal to 5 or 10, the power of the test is really good, since the proportion of rejections is close to one, in the majority of the cases. Again, this proportion depends on the value of the bandwidth.