Goodness-of-fit tests for the spatial spectral density

Abstract

Detection and modeling the spatial correlation is an important issue in spatial data analysis. We extend in this work two different goodness-of-fit testing techniques for the spatial spectral density. The first approach is based on a smoothed version of the ratio between the periodogram and a parametric estimator of the spectral density. The second one is a generalized likelihood ratio test statistic, based on the log-periodogram representation as the response variable in a regression model. As a particular case, we provide tests for independence. Asymptotic normal distribution of both statistics is obtained, under the null hypothesis. For the application in practice, a resampling procedure for calibrating these tests is also given. The performance of the method is checked by a simulation study. Application to real data is also provided.

Appendix

For the sake of simplicity, we just give the sketch of the proofs for the results in these work. Detailed proofs can be found in Crujeiras et al. (2007).

Lemma 1

Assume that assumption (2) is fulfilled and considerU _k independent identically distributed random variables withE(U _k ) = 1, Var(U _k ) = 1 andE(U ⁴ _k ) < ∞. Then,

$$ {\frac{|H|^{1/4}}{N}}\int\limits_{\Uppi^2}\left(\sum_{{\mathbf{k}}} K_H({\varvec{\lambda}}-{\varvec{\lambda}}_{{\mathbf{k}}}) (U_{{\mathbf{k}}}-1)\right)^2d{\varvec{\lambda}}-\mu_H\rightarrow N(0,\tau^2), $$

where μ _H and τ² are given in (22) and (23), respectively and the sum$\sum_{{\mathbf{k}}}$extends over the set of Fourier frequencies.

Proof

See Crujeiras et al. (2007) for detailed proof. $\square$

Lemma 2

LetT ⁰ _P denote the test statistic in (14) replacing$\hat\theta$by the true parameter θ₀. Then, under assumptions in Theorem 1:

$$ T_P=T_P^0+o_{{\mathbb{P}}}(1). $$

Proof

This result is proved by similar reasoning to Lemma 7 in Paparoditis (2000). $\square$

Lemma 3

If θ = θ₀ is the true parameter, under assumptions (1–4):

$$ T_P^0-\mu_H\rightarrow N(0,\tau^2), $$

asN → ∞, where μ _H and τ² are given in (22) and (23), respectively and T ⁰ _P is given by (14) replacing$\hat\theta$by the true parameter θ₀.

Proof

Denote by W _k  = 1−V _k , where V _k are the iid standard exponential random variables in the periodogram expression (6), with $f=f_{\theta_{0}}$. The statistic T ⁰ _P can be decomposed in three addends in the following way:

$$ \begin{aligned} &T_P^0-\mu_H={\frac{|H|^{1/4}}{N}}\int\limits_{\Uppi^2}\left( \sum_{{{\mathbf{k}}}}K_H({\varvec{\lambda}}-{\varvec{\lambda}} _{{\mathbf{k}}})W_{{\mathbf{k}}}\right)^2d{\varvec{\lambda}}-\mu_H +{\frac{|H|^{1/4}}{N}}\int\limits_{\Uppi^2}\left(\sum_{{{\mathbf{k}}}} K_H({\varvec{\lambda}}-{\varvec{\lambda}}_{{\mathbf{k}}}) {\frac{R_N({\varvec{\lambda}}_{{{\mathbf{k}}}})}{f_{\theta_0} ({\varvec{\lambda}}_{{{\mathbf{k}}}})}}\right)^2d{\varvec{\lambda}}\\ &\quad+{\frac{2|H|^{1/4}}{N}}\int\limits_{\Uppi^2}\sum_{{{\mathbf{k}}}}\sum_{{{\mathbf{j}}}} K_H({\varvec{\lambda}}-{\varvec{\lambda}}_{{{\mathbf{k}}}})K_H ({\varvec{\lambda}}-{\varvec{\lambda}}_{{{\mathbf{j}}}})W_{{\mathbf{k}}} {\frac{R_N({\varvec{\lambda}}_{{{\mathbf{j}}}})} {f_{\theta_0}({\varvec{\lambda}}_{{{\mathbf{j}}}})}}d{\varvec{\lambda}} \end{aligned} $$

Following similar results to those in the proof of Lemma 5 in Paparoditis (2000), the first addend tends to zero in probability. The same conclusion holds for the third adend, as an extension of Lemma 4 in Paparoditis (2000). The result is proved by Lemma 1. $\square$

Proof of Theorem 1

Theorem 1 is proved combining the results in Lemmas 2 and 3. $\square$

Lemma 4

Under assumptions (2) and (5), if f is bounded and bounded away from zero, then

$$ \sqrt{N}(\hat\theta-\theta^*)-\sqrt{N}\int\limits_{\Uppi^2}W ({\varvec{\lambda}})(I({\varvec{\lambda}})-f({\varvec{\lambda}})) d{\varvec{\lambda}}\rightarrow 0 $$

in probability, where

$$ W({\varvec{\lambda}})=-{{\mathcal{H}}}^{-1}\nabla f_{\theta}^{-1}({\varvec{\lambda}})|_{\theta=\theta^*},\quad \quad {{\mathcal{H}}}=\int\limits_{\Uppi^2}\nabla^2G(\theta^*,f,{\varvec{\lambda}}) d{\varvec{\lambda}},\quad G(\theta,f,{\varvec{\lambda}})=\log f_{\theta}({\varvec{\lambda}})+{\frac{f({\varvec{\lambda}})} {f_\theta({\varvec{\lambda}})}}. $$

Proof

The prove of this lemma is obtained generalizing Theorem 3.2 in Dahlhaus and Wefelmeyer (1996). $\square$

Proof of Theorem 2

Once we have obtained the $\sqrt{N}$-consistency of $\hat\theta$ as an estimator of θ*, the proof of the theorem is analogous as the proof of Theorem 3 in Paparoditis (2000). $\square$

Proof of Theorem 3

The Generalized Likelihood Ratio Test statistic (18) can be decomposed as:

$$ T_{LK}=T_{LK}^*+B_1+B_2-B_3 $$

(33)

where T ^* _LK is the same as T _LK but replacing Y _k by Y ^** _k and ${\hat{m}}_{LK}({\varvec{\lambda}}_{{\mathbf{k}}})$ by ${\hat{m}}_{LK}^*({\varvec{\lambda}}_{{\mathbf{k}}})$ and,

$$\begin{aligned}&B_1=\sum_{{\mathbf{k}}}\left\{1-e^{Y_{{\mathbf{k}}}-{\hat{m}}_{LK}^* ({\varvec{\lambda}}_{{\mathbf{k}}})}\right\}\left({\hat{m}}_{LK} ({\varvec{\lambda}}_{{\mathbf{k}}})-{\hat{m}}_{LK}^*({\varvec{\lambda}} _{{\mathbf{k}}})\right),\\&B_2=\sum_{{\mathbf{k}}} e^{Y_{{\mathbf{k}}}-{\hat{m}}_{LK}^*({\varvec{\lambda}}_{{\mathbf{k}}})} \left({\hat{m}}_{LK}({\varvec{\lambda}}_{{\mathbf{k}}})-{\hat{m}}_{LK}^* ({\varvec{\lambda}}_{{\mathbf{k}}})\right)^2,\\&B_3=\sum_{{\mathbf{k}}} {\frac{R_N({\varvec{\lambda}}_{{\mathbf{k}}})} {f_{\theta}({\varvec{\lambda}}_{{\mathbf{k}}})}}\left\{e^{m_{\theta} ({\varvec{\lambda}}_{{\mathbf{k}}})-{\hat{m}}_{LK}^*({\varvec{\lambda}} _{{\mathbf{k}}})}-1\right\}.\end{aligned}$$

Theorem 3 is proved considering the extension of Theorem 10 in Fan et al. (2001) to the bidimensional case and proceeding similarly to Fan and Zhang (2004), in order to bound B ₁, B ₂ and B ₃. Asymptotic normality of the T ^* _LK statistics is obtained decomposing this statistic as:

$$ T_{LK}^*=\mu_H+R_N+{\frac{1}{2}}W_N|H|^{-1/4}, $$

where R _N is negligible and $W_N=\sum_{{{\mathbf{i}}}}\sum_{{{\mathbf{j}}}}b({{\mathbf{i}}}, {{\mathbf{j}}})\varepsilon_{{{\mathbf{i}}}}\varepsilon_{{{\mathbf{j}}}},$ with $\varepsilon_{{{\mathbf{k}}}}=e^{Y_{{{\mathbf{k}}}}^{**}-m_{\theta} ({\varvec{\lambda}}_{{{\mathbf{k}}}})}-1$ and Y ^** _k given in (15). In order to prove the asymptotic normal distribution of W _N , we apply Proposition 3.2 in de Jong (1987). For that purpose, we write W _N as a quadratic form of independent random variables, as in the development of Lemma 1. $\square$