A computational validation for nonparametric assessment of spatial trends

Abstract

The analysis of continuously spatially varying processes usually considers two sources of variation, namely, the large-scale variation collected by the trend of the process, and the small-scale variation. Parametric trend models on latitude and longitude are easy to fit and to interpret. However, the use of parametric models for characterizing spatially varying processes may lead to misspecification problems if the model is not appropriate. Recently, Meilán-Vila et al. (TEST 29:728–749, 2020) proposed a goodness-of-fit test based on an $L_2$-distance for assessing a parametric trend model with correlated errors, under random design, comparing parametric and nonparametric trend estimates. The present work aims to provide a detailed computational analysis of the behavior of this approach using different bootstrap algorithms for calibration, one of them including a procedure that corrects the bias introduced by the direct use of the residuals in the variogram estimation, under a fixed design geostatistical framework. Asymptotic results for the test are provided and an extensive simulation study, considering complexities that usually arise in geostatistics, is carried out to illustrate the performance of the proposal. Specifically, we analyze the impact of the sample size, the spatial dependence range and the nugget effect on the empirical calibration and power of the test.

Appendix. Proof of the main theorem

In this appendix, under assumptions (A1)–(A8), Theorem 1 is proved. The asymptotic distribution of the test statistic, given in (7), is derived. This test compares the nonparametric and the smooth parametric estimators, given in (6) and (8), respectively, using an $L_2$-distance.

Proof

The test statistic (7) can be decomposed as

$$\begin{aligned} T_n= & {} n|\varvec{H}|^{1/2}\int _{}^{}\left[ \hat{m}^{NW}_{\varvec{H}}(\varvec{s})-\hat{m}^{NW}_{\varvec{H},\hat{{\varvec{\beta }}}}(\varvec{s})\right] ^2w(\varvec{s})d\varvec{s}\\= & {} n|\varvec{H}|^{1/2}\int _{}^{}\left[ \dfrac{\sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})Z_i}{\sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})}-\dfrac{\sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})m_{\hat{{{\varvec{\beta }}}}}(\varvec{s}_i)}{\sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})}\right] ^2w(\varvec{s})d\varvec{s}\\= & {} n|\varvec{H}|^{1/2}\int _{}^{}\dfrac{\left\{ \sum _{i=1}^n K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) \left[ m(\varvec{s}_i)+\varepsilon _i-m_{\hat{{{\varvec{\beta }}}}}(\varvec{s}_i)\right] \right\} ^2}{\left[ \sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})\right] ^2}w(\varvec{s})d\varvec{s}. \end{aligned}$$

Now, taking into account that the trends considered are of the form $m=m_{{{\varvec{\beta }}}_0}+ n^{-1/2}|\varvec{H}|^{-1/4}g$, one gets:

$$\begin{aligned} T_n= & {} n|\varvec{H}|^{1/2}\int _{}^{}\dfrac{\left\{ \sum _{i=1}^n K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) \left[ m_{{{\varvec{\beta }}}_0}(\varvec{s}_i)+n^{-1/2}|\varvec{H}|^{-1/4}g(\varvec{s}_i)+\varepsilon _i-m_{\hat{{{\varvec{\beta }}}}}(\varvec{s}_i)\right] \right\} ^2}{\left[ \sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})\right] ^2}w(\varvec{s})d\varvec{s} \\ {}= & {} n|\varvec{H}|^{1/2}\int _{}^{}\left[ I_1(\varvec{s})+I_2(\varvec{s})+I_3(\varvec{s})\right] ^2w(\varvec{s})d\varvec{s}, \end{aligned}$$

where

$$\begin{aligned} I_1(\varvec{s})= & {} \dfrac{\sum _{i=1}^n K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) \left[ m_{{{\varvec{\beta }}}_0}(\varvec{s}_i)-m_{\hat{{{\varvec{\beta }}}}}(\varvec{s}_i)\right] }{\sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})},\\ I_2(\varvec{s})= & {} \dfrac{\sum _{i=1}^n K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) n^{-1/2}|\varvec{H}|^{-1/4}g(\varvec{s}_i)}{\sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})},\\ I_3(\varvec{s})= & {} \dfrac{\sum _{i=1}^n K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) \varepsilon _i}{\sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})}. \end{aligned}$$

Under assumptions (A1), (A2) and (A6), and given that the difference $m_{\hat{{{\varvec{\beta }}}}}(\varvec{s})-m_{{{{\varvec{\beta }}}}_0}(\varvec{s})=O_p(n^{-1/2})$, it is obtained that

$$\begin{aligned} n|\varvec{H}|^{1/2}\int _{}^{}I_1^2(\varvec{s})w(\varvec{s})d\varvec{s}= & {} n|\varvec{H}|^{1/2}\int _{}^{}\left\{ \dfrac{\sum _{i=1}^n K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) \left[ m_{{{\varvec{\beta }}}_0}(\varvec{s}_i)-m_{\hat{{{\varvec{\beta }}}}}(\varvec{s}_i)\right] }{\sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})}\right\} ^2w(\varvec{s})d\varvec{s}\\= & {} O_p(|\varvec{H}|^{1/2}). \end{aligned}$$

For the term $I_2(\varvec{s})$, using the assumption (A2), it follows that

$$\begin{aligned} n|\varvec{H}|^{1/2}\int _{}^{}I_2^2(\varvec{s})w(\varvec{s})d\varvec{s}= & {} n|\varvec{H}|^{1/2}\int _{}^{}\left[ \dfrac{\sum _{i=1}^n K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) n^{-1/2}|\varvec{H}|^{-1/4}g(\varvec{s}_i)}{\sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})}\right] ^2w(\varvec{s})d\varvec{s}\nonumber \\ {}= & {} n|\varvec{H}|^{1/2}n^{-1}|\varvec{H}|^{-1/2}\int _{}^{} \left[ {K}_{\varvec{H}}*g(\varvec{s})\right] ^2w(\varvec{s})d\varvec{s}\nonumber \\ {}= & {} \int _{}^{} \left[ {K}_{\varvec{H}}*g(\varvec{s})\right] ^2w(\varvec{s})d\varvec{s}, \end{aligned}$$

(12)

which corresponds to $b_{1\varvec{H}}$ in Theorem 1. Finally, $I_3(\varvec{s})$ (associated with the error component) can be decomposed as:

$$\begin{aligned} n|\varvec{H}|^{1/2}\int _{}^{}I_3^2(\varvec{s})w(\varvec{s})d\varvec{s}= & {} n|\varvec{H}|^{1/2}\int _{}^{}\left[ \dfrac{\sum _{i=1}^n K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) \varepsilon _i}{\sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})}\right] ^2w(\varvec{s})d\varvec{s}\\ {}= & {} n|\varvec{H}|^{1/2}\int _{}^{}\dfrac{\sum _{i=1}^n K^2_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) \varepsilon ^2_i}{\left[ \sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})\right] ^2}w(\varvec{s})d\varvec{s} \\&+ n|\varvec{H}|^{1/2}\int _{}^{}\dfrac{\sum _{i\ne j} K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) K_{\varvec{H}}\left( {\varvec{s}_j-\varvec{s}}\right) \varepsilon _i\varepsilon _j}{\left[ \sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})\right] ^2}w(\varvec{s})d\varvec{s} \\= & {} I_{31}+I_{32}. \end{aligned}$$

Close expressions of $I_{31}$ and $I_{32}$ can be obtained computing the expectation and the variance of these terms. Under assumption (A6), it can be proved that

$$\begin{aligned} \mathbb {E}(|\varvec{H}|^{1/2}I_{31})= & {} \mathbb {E}\left\{ n|\varvec{H}|\int _{}^{}\dfrac{\sum _{i=1}^n K^2_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) \varepsilon ^2_i}{\left[ \sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})\right] ^2}w(\varvec{s})d\varvec{s}\right\} \nonumber \\ {}= & {} {}\sigma ^2K^{(2)}(0)\int _{}^{}w(\varvec{s})d\varvec{s}\cdot \left[ 1+o(1)\right] . \end{aligned}$$

(13)

Similarly, using assumptions (A3), (A6) and (A7), it can be obtained that

$$\begin{aligned} \text{ Var }(|\varvec{H}|^{1/2}I_{31})= & {} {\text{ Var }}\left\{ n|\varvec{H}|\int _{}^{}\dfrac{\sum _{i=1}^n K^2_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) \varepsilon ^2_i}{\left[ \sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})\right] ^2}w(\varvec{s})d\varvec{s}\right\} \\ {}= & {} 2n^2|\varvec{H}|^2\sum _{i=1}^n\sum _{j=1}^n\int _{}^{}\int _{}^{}\dfrac{K^2_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) K^2_{\varvec{H}}\left( {\varvec{s}_j-\varvec{t}}\right) [{\text{ Cov }}(\varepsilon _i,\varepsilon _j)]^2}{\left[ \sum _{i=1}^n K^2_{\varvec{H}}({\varvec{s}_i-\varvec{s}})\right] \left[ \sum _{i=1}^n K^2_{\varvec{H}}({\varvec{s}_i-\varvec{t}})\right] }w(\varvec{s})w(\varvec{t})d\varvec{s}d\varvec{t}\\ {}= & {} 2\sigma ^4|\varvec{H}|\int _{}^{}\int _{}^{}\int _{}^{}\int _{}^{}K^2(\varvec{v})K^2(\varvec{z})w^2(\varvec{s})\rho _n^2[\varvec{H}(\varvec{v}-\varvec{z}+\varvec{u})]d\varvec{v}d\varvec{z}d\varvec{s}d\varvec{u}\cdot \left[ 1+o(1)\right] .\\ \end{aligned}$$

Let

$$\begin{aligned} j_n(\varvec{v},\varvec{u})= & {} n|\varvec{H}|\int K^2(\varvec{v})\rho ^2_n[\varvec{H}(\varvec{v}-\varvec{z}+\varvec{u})]d\varvec{z}. \end{aligned}$$

Notice that, using assumption (A3),

$$\begin{aligned} |j_n(\varvec{v},\varvec{u})|\le & {} K^2_M\left\{ n|\varvec{H}|\int |\rho ^2_n[\varvec{H}(\varvec{v}-\varvec{z}+\varvec{u})]|d\varvec{z}\right\} \\ {}\le & {} K^2_M\left[ n\int |\rho _n(\varvec{t})|d\varvec{t}\right] \\ {}\le & {} K_M^2\rho _{M}, \end{aligned}$$

where $K_M {=} \max _{\varvec{s}}[K(\varvec{s})]$ and $\rho _M {=} \max _{\varvec{s}}[\rho _n(\varvec{s})]$, and using assumptions (A2), (A3), (A6) and (A8), one gets that

$$\begin{aligned} \text{ Var }(I_{31})= & {} o(1). \end{aligned}$$

(14)

From (13) and (14) it follows that

$$\begin{aligned} I_{31}=\sigma ^2 |\varvec{H}|^{-1/2}{}K^{(2)}(\varvec{0})\int w(\varvec{s})d\varvec{s}\cdot \left[ 1+o_p(1)\right] . \end{aligned}$$

(15)

Now, consider the term

$$\begin{aligned} I_{32}= & {} n|\varvec{H}|^{1/2}\int _{}^{}\dfrac{\sum _{i\ne j} K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) K_{\varvec{H}}\left( {\varvec{s}_j-\varvec{s}}\right) \varepsilon _i\varepsilon _j}{\left[ \sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})\right] ^2}w(\varvec{s})d\varvec{s}.\\ \end{aligned}$$

Let

$$\begin{aligned} \kappa _{ij}= & {} n|\varvec{H}|^{1/2}\displaystyle \int _{}^{}\dfrac{K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) K_{\varvec{H}}\left( {\varvec{s}_j-\varvec{s}}\right) }{\left[ \sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})\right] ^2}w(\varvec{s})d\varvec{s}{\varepsilon }(\varvec{s}_i){\varepsilon }(\varvec{s}_j). \end{aligned}$$

Thus,

$$\begin{aligned} I_{32}=\sum _{i\ne j}\kappa _{ij}, \end{aligned}$$

and this can be seen as a U-statistic with degenerate kernel. To obtain the asymptotic normality of $I_{32}$ we apply the central limit theorem for reduced U-statistics under dependence given by Kim et al. (2013).

For this term $I_{32}$ we have

$$\begin{aligned} \mathbb {E}\left( |\varvec{H}|^{1/2}I_{32}\right)= & {} \mathbb {E}\left\{ n|\varvec{H}|\int _{}^{}\dfrac{\sum _{i\ne j} K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) K_{\varvec{H}}\left( {\varvec{s}_j-\varvec{s}}\right) \varepsilon _i\varepsilon _j}{\left[ \sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})\right] ^2}w(\varvec{s})d\varvec{s} \right\} \\ {}= & {} n^{-1}|\varvec{H}|\sum _{i\ne j}\text{ Cov }[{\varepsilon }(\varvec{s}_i),{\varepsilon }(\varvec{s}_j)]\displaystyle \int _{}^{}\dfrac{K_{\varvec{H}}\left( {\varvec{s}_i-\varvec{s}}\right) K_{\varvec{H}}\left( {\varvec{s}_j-\varvec{s}}\right) }{\left[ \sum _{i=1}^n K_{\varvec{H}}({\varvec{s}_i-\varvec{s}})\right] ^2}w(\varvec{s})d\varvec{s}\cdot \left[ 1+o(1)\right] \\ {}= & {} (n-1)|\varvec{H}|\sigma ^2\displaystyle \int _{}^{}\int _{}^{}\int _{}^{}{K\left( \varvec{v}\right) K\left( \varvec{z}\right) }\rho _n[\varvec{H}(\varvec{v}-\varvec{z})]w(\varvec{s})d\varvec{v}d\varvec{z}d\varvec{s}\cdot \left[ 1+o(1)\right] .\end{aligned}$$

Under the assumptions (A4)–(A8), as shown by Liu (2001),

$$\begin{aligned} \lim _{n\rightarrow \infty }n|\varvec{H}|\int K(\varvec{v})K(\varvec{z})\rho _n[\varvec{H}(\varvec{v}-\varvec{z})]d\varvec{v}d\varvec{z}=K^{(2)}(0)\rho _c. \end{aligned}$$

It follows that

$$\begin{aligned} \mathbb {E}(|\varvec{H}|^{1/2}I_{32})=\sigma ^2K^{(2)}(0)\rho _{c}\int _{}^{}w(\varvec{s})d\varvec{s}\cdot \left[ 1+o(1)\right] . \end{aligned}$$

(16)

Similarly, it can be obtained that the asymptotic variance of $I_{32}$ is

$$\begin{aligned} V=\sigma ^4 K^{(4)}(0)\int _{}^{} w^2(\varvec{s})d\varvec{s}\left( 1+\rho _{c}+2\rho ^2_c\right) . \end{aligned}$$

(17)

The term $I_{32}$ converges in distribution to a normally distributed random variable with mean the second term of $b_{0\varvec{H}}$ and variance V.

In virtue of the Cauchy–Bunyakovsky–Schwarz inequality, the cross terms in $T_n$ resulting from the products of $I_1$, $I_2$ and $I_3$ are all of small order. Therefore, combining the results given in Eqs. (12) and (15), and the asymptotic normality of $I_{32}$ (with its bias (16) and its variance (17)), it follows that

$$\begin{aligned} V^{-1/2}(T_n-b_{0\varvec{H}}-b_{1\varvec{H}})\rightarrow _{\mathcal {L}} N(0,1), \text { as } n\rightarrow \infty , \end{aligned}$$

where

$$\begin{aligned} b_{0\varvec{H}}= & {} |\varvec{H}|^{-1/2}\sigma ^2K^{(2)}(0)\int _{}^{}{w(\varvec{s})}d\varvec{s}\left( 1+\rho _{c}\right) ,\\ b_{1\varvec{H}}= & {} \int _{}^{}\left[ {K}_{\varvec{H}}*g(\varvec{s})\right] ^2w(\varvec{s})d\varvec{s},\\ V= & {} \sigma ^4 K^{(4)}(0)\int _{}^{} w^2(\varvec{s})d\varvec{s}\left( 1+\rho _{c}+2\rho ^2_c\right) . \end{aligned}$$

$\square$