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Bootstrap approaches for spatial data

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Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript

Abstract

Generation of replicates of the available data enables the researchers to solve different statistical problems, such as the estimation of standard errors, the inference of parameters or even the approximation of distribution functions. With this aim, Bootstrap approaches are suggested in the current work, specifically designed for their application to spatial data, as they take into account the dependence structure of the underlying random process. The key idea is to construct nonparametric distribution estimators, adapted to the spatial setting, which are distribution functions themselves, associated to discrete or continuous random variables. Then, the Bootstrap samples are obtained by drawing at random from the estimated distribution. Consistency of the suggested approaches will be proved by assuming stationarity from the random process or by relaxing the latter hypothesis to admit a deterministic trend. Numerical studies for simulated data and a real data set, obtained from environmental monitoring, are included to illustrate the application of the proposed Bootstrap methods.

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Acknowledgments

The authors would like to thank the helpful suggestions and comments from the Reviewers. The authors are also grateful to Dr. K. J. Duncan-Barlow (University of Vigo) for her contribution in the language revision. The first and third authors acknowledge financial support from the Project TEC2011-28683-C02-02 of the Spanish Ministry of Science and Innovation and the Project CN2012/279 from the European Regional Development Fund and the Galician Regional Government (Xunta de Galicia). The second author’s work has been supported by the Project PTDC/MAT/112338/2009 (FEDER support included) of the Portuguese Ministry of Science, Technology and Higher Education.

Author information

Authors and Affiliations

  1. Department of Statistics and Operations Research, University of Vigo, Vigo, Spain

    Pilar García-Soidán

  2. Department of Mathematics and Applications, University of Minho, Braga, Portugal

    Raquel Menezes

  3. Department of Signal Theory and Communications, University of Vigo, Vigo, Spain

    Óscar Rubiños

Authors
  1. Pilar García-Soidán
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  2. Raquel Menezes
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  3. Óscar Rubiños
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Corresponding author

Correspondence to Pilar García-Soidán.

Appendices

Appendix 1: Consistency of \(\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right)\)

To check that consistency follows for \(\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right), \) the hypotheses described below will be assumed:

  1. (i)

    \(\{ Z ( {\rm s} ) \in {I\!R} : {\rm s} \in D \subset {I\!R}^d \}\) can be modeled as given in (1).

  2. (ii)

    D = λD 0, for some \(\lambda=\lambda (n) \mathop{\longrightarrow}\limits^{n \rightarrow + \infty} +\infty\) and bounded \(D_0 \subset {I\!R}^d. \)

  3. (iii)

    t i  = λu i , for 1 ≤ i ≤ n, where \({\rm u}_1, {\ldots}, {\rm u}_n\) denotes a realization of a random sample of size n drawn from a density function g 0 considered on D 0.

  4. (iv)

    \(Z(\cdot)\) is α-mixing, with α(r) = O(r −a), for r > 0 and some constant a > 0.

  5. (v)

    K is d-variate and symmetric density function with compact support.

  6. (vi)

    \(\{ h_{1}^{2}+ {\cdots} + h_{k-1}^{2}+ \lambda^{-1} + n^{-k} \lambda^{d(k-1)} h_{1}^{-d} {\ldots} h_{k-1}^{-d} \} \mathop{\longrightarrow}\limits^{n \rightarrow + \infty} 0. \)

  7. (vii)

    \(F_{{\rm s}_1,{\ldots}, {\rm s}_k} (x_1,{\ldots},x_k)\) is three-times continuously differentiable as a function of \(({\rm s}_1,{\ldots}, {\rm s}_k). \)

We will prove that the bias and the variance of \(\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right)\) are of the respective orders \(( h_{1}^{2}+ {\cdots} + h_{k-1}^{2})\) and \((n^{-k} \lambda^{d(k-1)} h_{1}^{-d} {\ldots} h_{k-1}^{-d} + \lambda^{-d})\) and, therefore, tend to zero as the sample size n increases, which would yield the consistency of the distribution estimator. To do the latter, conditions (i)–(vii) will be applied and a similar procedure as in the proof of Theorem 3.1 in Hall and Patil (1994).

Write \(A_{i_1,i_2}=\frac{ p_{{\rm s}_1,{\rm s}_2}^{{\rm t}_{i_1},{\rm t}_{i_2},h_1} }{\sum_{i_1=1}^{n} \sum_{i_2=1}^{n} p_{{\rm s}_1,{\rm s}_2}^{{\rm t}_{i_1},{\rm t}_{i_2},h_1}}\) and \(A_{i_{j-1},i_j}=\frac{ p_{{\rm s}_{j-1},{\rm s}_j}^{{\rm t}_{i_{j-1}},{\rm t}_{i_j},h_{j-1}} }{\sum_{i_j=1}^{n} p_{{\rm s}_{j-1},{\rm s}_j}^{{\rm t}_{i_{j-1}},{\rm t}_{i_j},h_{j-1}}}\) for \(j=3,{\ldots},k. \) Firstly, we can take into account that, for large n:

$$ {\rm E} \left[\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right)\right] ={\rm E} \left[ {\rm E} \left[ \hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right) {\rm t}_{i_j}, \forall j \right] \right] =\sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_k=1}^{n} {\rm E} \left[A_{i_1,i_2} {\ldots} A_{i_{k-1},i_k} {\rm E} \left[ I_{\{ X({\rm t}_{i_1}) \leq x_1 \}} {\ldots} I_{\{ X({\rm t}_{i_k}) \leq x_k \}} {\rm t}_{i_j}, \forall j \right]\right] = \sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_k=1}^{n} {\rm E} \left[ A_{i_1,i_2} {\ldots} A_{i_{k-1},i_k} F_{{\rm t}_{i_1},{\ldots}, {\rm t}_{i_k}} \left( x_1 + \mu \left( {\rm s}_1 + {\rm t}_{i_1}- {\rm s}_1\right) - \mu \left( {\rm s}_1 \right),{\ldots},x_k + \mu \left( {\rm s}_k + {\rm t}_{i_1}- {\rm s}_1\right) - \mu \left( {\rm s}_k \right)\right)\right] =\sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_k=1}^{n} {\rm E} \left[ A_{i_1,i_2} {\ldots} A_{i_{k-2},i_{k-1}} \cdot {\rm E} \left[ A_{i_{k-1},i_k} F_{{\rm t}_{i_1}-{\rm t}_{i_1}+{\rm s}_1, {\ldots},{\rm t}_{i_k}-{\rm t}_{i_1}+{\rm s}_1 } \left( x_1,{\ldots},x_k \right){\rm t}_{i_j}, j\leq k-1 \right] \right]$$

on account of (3).

Now, the last conditional expectation will be approximated. With this aim, bear in mind that:

$$ {\rm E} \left[ K\left( \frac{{\rm s_{k-1}}-{\rm s}_k-({\rm t}_{i_{k-1}}-{\rm t}_{i_k})}{h_{k-1}} \right) F_{{\rm t}_{i_1}-{\rm t}_{i_1}+{\rm s}_1, {\ldots},{\rm t}_{i_k}-{\rm t}_{i_1}+{\rm s}_1 } \left( x_1,{\ldots},x_k \right)\right]= \int K\left( \frac{{\rm s_{k-1}}-{\rm s}_k-({\rm t}_{i_{k-1}}-\lambda {\rm u})}{h_{k-1}} \right) F_{{\rm t}_{i_1}-{\rm t}_{i_1}+{\rm s}_1, {\ldots},\lambda {\rm u}-{\rm t}_{i_1}+{\rm s}_1 } \left( x_1,{\ldots},x_k \right) g_0({\rm u}) d{\rm u}\approx \lambda^d h_{k-1}^{d} g_0(0) \int K\left( {\rm z}_1 \right) F_{{\rm t}_{i_1}-{\rm t}_{i_1}+{\rm s}_1, {\ldots},{\rm s}_k-{\rm s}_{k-1}+{\rm t}_{i_{k-1}} -{\rm t}_{i_1}+{\rm s}_1+h_{k-1} {\rm z}_1 } \left( x_1,{\ldots},x_k \right) d{\rm z}_1 {\rm E} \left[ K\left( \frac{{\rm s_{k-1}}-{\rm s}_k-({\rm t}_{i_{k-1}}-{\rm t}_{i_k})}{h_{k-1}} \right) \right] = \int K\left( \frac{{\rm s_{k-1}}-{\rm s}_k-({\rm t}_{i_{k-1}}-\lambda {\rm u})}{h_{k-1}} \right) g_0({\rm u}) d{\rm u} \approx \lambda^d h_{k-1}^{d} g_0(0) $$

From the previous relations, it follows that:

$${\rm E} \left[ \left. A_{i_{k-1},i_k} F_{{\rm t}_{i_1}-{\rm t}_{i_1}+{\rm s}_1, {\ldots},{\rm t}_{i_k}-{\rm t}_{i_1}+{\rm s}_1 } \left( x_1,{\ldots},x_k \right)\right/ {\rm t}_{i_j}, j\leq k-1 \right]\approx n^{-1} \int K\left( {\rm z}_1 \right) F_{{\rm t}_{i_1}-{\rm t}_{i_1}+{\rm s}_1, {\ldots},{\rm s}_k-{\rm s}_{k-1}+{\rm t}_{i_{k-1}} -{\rm t}_{i_1}+{\rm s}_1+h_{k-1}{\rm z}_1 } \left( x_1,{\ldots},x_k \right) d{\rm z}_1 $$

and, therefore:

$${\rm E} \left[\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right)\right] \approx \sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_{k-1}=1}^{n} {\rm E} \left[ A_{i_1,i_2} {\ldots} A_{i_{k-2},i_{k-1}} \int K\left( {\rm z}_1 \right) \cdot F_{{\rm t}_{i_1}-{\rm t}_{i_1}+{\rm s}_1, {\ldots},{\rm s}_k-{\rm s}_{k-1}-{\rm t}_{i_{k-1}} -{\rm t}_{i_1}+{\rm s}_1 +h_{k-1}{\rm z}_1} \left( x_1,{\ldots},x_k \right) d{\rm z}_1 \right].$$

We can iterate the strategy above, based on applying an appropriate conditional expectation and developing the resulting term, to achieve that:

$${\rm E} \left[\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right)\right]\approx \int {\ldots} \int F_{{\rm s}_1,{\rm s}_2+h_1{\rm z}_{k-1},{\ldots}, {\rm s}_k+h_1{\rm z}_{k-1}+ {\cdots} +h_{k-1}{\rm z}_1} \left( x_1,x_2,{\ldots},x_k \right) \cdot K \left( {\rm z}_1 \right) {\ldots} K \left( {\rm z}_{k-1} \right) d{\rm z}_1 {\ldots} d{\rm z}_{k-1} = F_{{\rm s}_1,{\rm s}_2,{\ldots}, {\rm s}_k} \left( x_1,x_2,{\ldots},x_k \right) + O \left( h_{1}^{2}+{\cdots}+ h_{k-1}^{2}\right) $$

With regard to the variance, one has for large n that:

$$ {\rm Var} \left[\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right)\right]={\rm E} \left[ \left(\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right) -{\rm E} \left[\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right)\right]\right)^2 \right] \approx V_1+V_2 $$

where:

$$ \begin{aligned} V_1 &= \sum\limits_{i_1=1}^{n} {\ldots}\sum\limits_{i_k=1}^{n} {\rm E} \left[ A_{i_1,i_2}^{2} {\ldots}A_{i_{k-1},i_k}^{2}\cdot \left( I_{\{ X({\rm t}_{i_1}) \leq x_1\}} {\ldots} I_{\{ X({\rm t}_{i_k}) \leq x_k \}} - F_{{\rm s}_1,{\ldots},{\rm s}_k } \left( x_1,{\ldots},x_k \right)^2 \right)\right]\\ V_2 &= \sum\limits_{i_1=1}^{n} {\ldots}\sum\limits_{i_k=1}^{n} \sum\limits_{j_1=1}^{n} {\ldots}\sum\limits_{j_k=1}^{n}\cdot{\rm E} \left[ A_{i_1,i_2} {\ldots}A_{i_{k-1},i_k} A_{j_1,j_2} {\ldots} A_{j_{k-1},j_k}\left( I_{\{X({\rm t}_{i_1}) \leq x_1 \}} {\ldots} I_{\{ X({\rm t}_{i_k}) \leq x_k \}} \cdot I_{\{ X({\rm t}_{j_1}) \leq x_1 \}} {\ldots} I_{\{X({\rm t}_{j_k}) \leq x_k \}} -F_{{\rm s}_1, {\ldots},{\rm s}_k }\left( x_1,{\ldots},x_k \right)^2 \right) \right] \end{aligned}$$

By using similar arguments as above, we could check that:

$$ V_1 \approx \frac{n^{-k} \lambda^{d(k-1)} h_{1}^{-d} {\ldots} h_{k-1}^{-d} \left( F_{{\rm s}_1,{\ldots}, {\rm s}_k} \left( x_1,{\ldots},x_k \right)- F_{{\rm s}_1,{\ldots}, {\rm s}_k} \left( x_1,{\ldots},x_k \right)^2 \right)}{ g_0(0)^{k}\left( \int K \left( {\rm z} \right)^2 d{\rm z} \right)^{k-1} } V_2 \approx \lambda^{-d} \int \left( F_{{\rm s}_1,{\ldots}, {\rm s}_k,{\rm s}_1+{\rm t},{\ldots}, {\rm s}_k+{\rm t}} \left( x_1,{\ldots},x_k ,x_1,{\ldots},x_k\right) - F_{{\rm s}_1,{\ldots}, {\rm s}_k} \left( x_1,{\ldots},x_k \right)^2 \right) d{\rm t} $$

Consequently:

$$ {\rm Var} \left[\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right)\right]= O \left( n^{-k} \lambda^{d(k-1)} h_{1}^{-d} {\ldots} h_{k-1}^{-d} + \lambda^{-d} \right) $$

We could derive the dominant terms of the bias and the variance of the distribution estimator as well as asymptotically minimize the mean squared error (MSE) of the distribution estimator, namely:

$${\rm MSE} \left[\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right) \right]={\rm Bias} \left[ \hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right) \right]^2 + {\rm Var} \left[\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)} \left( x_1,{\ldots},x_k \right)\right] $$

to obtain the optimal bandwidths h j , for \(j=1,{\ldots},k-1, \) which would be dependent on unknown terms, such as the multivariate distribution function itself and its second-order derivatives.

Appendix 2: Consistency of \(\tilde{F}_{{\rm s}_1,{\ldots}, {\rm s}_k} \left( x_1,{\ldots},x_k \right)\)

To derive this proof, we will assume conditions (i)–(v), together with:

  1. (vi′)

    \(\{ h_{1}^{2}+ {\cdots} + h_{k-1}^{2}+ h^2+\lambda^{-1} + n^{-k} \lambda^{d(k-1)} h_{1}^{-d} {\ldots} h_{k-1}^{-d} \} \mathop{\longrightarrow}\limits^{n \rightarrow + \infty} 0. \)

  2. (vii′)

    \(F_{{\rm s}_1,{\ldots}, {\rm s}_k} (x_1,{\ldots},x_k)\) is three-times continuously differentiable as a function of \(({\rm s}_1,{\ldots}, {\rm s}_k)\) and as a function of \((x_1,{\ldots},x_k). \)

  3. (viii)

    L is a univariate and symmetric density function with compact support.

For large n, the aforementioned hypotheses yield that:

$$ {\rm E} \left[\tilde{F}_{{\rm s}_1,{\ldots}, {\rm s}_k} \left( x_1,{\ldots},x_k \right)\right] ={\rm E} \left[ {\rm E} \left[ \left. \tilde{F}_{{\rm s}_1,{\ldots}, {\rm s}_k} \left( x_1,{\ldots},x_k \right) \right/ {\rm t}_{i_j}, \forall j \right] \right] =\sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_k=1}^{n} {\rm E} \left[ \left. A_{i_1,i_2} {\ldots} A_{i_{k-1},i_{k}} {\rm E} \left[ {\cal L} \left( \frac{x_1 -X ( {\rm t}_{i_1} )}{h} \right) {\ldots} {\cal L} \left( \frac{x_k -X ( {\rm t}_{i_k} )}{h} \right) \right/ {\rm t}_{i_j}, \forall j \right]\right] =\sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_k=1}^{n} {\rm E} \left[ A_{i_1,i_2}\cdot{\ldots}\cdot A_{i_{k-1},i_{k}} \cdot \int {\cal L} \left( \frac{x_1 -u_1+\mu ( {\rm s}_1-{\rm t}_{i_1}+{\rm s}_1)-\mu({\rm s}_1)}{h} \right) {\ldots} {\cal L} \left( \frac{x_k -u_k+\mu ( {\rm s}_k-{\rm t}_{i_1}+{\rm s}_1)-\mu({\rm s}_k)}{h} \right) \cdot f_{{\rm t}_{i_1},{\ldots}, {\rm t}_{i_k}} \left( u_1,{\ldots},u_k\right) du_1 {\ldots} du_k \right]$$

where \(f_{{\rm t}_1,{\ldots}, {\rm t}_k}\) denotes the joint density function of \((Z({\rm t}_1),{\ldots},Z({\rm t}_k)). \)

We can integrate by parts and apply relation (3) to obtain that:

$$ {\rm E} \left[\tilde{F}_{{\rm s}_1,{\ldots}, {\rm s}_k} \left( x_1,{\ldots},x_k \right)\right] = \sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_k=1}^{n} {\rm E} \left[ A_{i_1,i_2} {\ldots} A_{i_{k-1},i_{k}} \int L\left( y_1\right) {\ldots} L\left( y_k\right) \cdot F_{{\rm t}_{i_1},{\ldots}, {\rm t}_{i_k}} \left( x_1 -hy_1+\mu ( {\rm s}_1-{\rm t}_{i_1}+{\rm s}_1)-\mu({\rm s}_1) , {\ldots}, x_k -hy_k+\mu ( {\rm s}_k-{\rm t}_{i_1}+{\rm s}_1)-\mu({\rm s}_k)\right) dy_1 {\ldots} dy_k \right]= \sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_k=1}^{n} {\rm E} \left[ A_{i_1,i_2} {\ldots} A_{i_{k-1},i_{k}} \int L\left( y_1\right) {\ldots} L\left( y_k\right) \cdot F_{{\rm t}_{i_1}-{\rm t}_{i_1}+{\rm s}_1, {\ldots},{\rm t}_{i_k}-{\rm t}_{i_1}+{\rm s}_1 } \left( x_1 -hy_1 , {\ldots}, x_k -hy_k\right) dy_1 {\ldots} dy_k \right] $$

By proceeding with analogue arguments as those used for the bias of \(\hat{F}_{{\rm s}_1,{\ldots}, {\rm s}_k}^{(2)}, \) it follows that:

$${\rm E} \left[\tilde{F}_{{\rm s}_1,{\ldots}, {\rm s}_k} \left( x_1,{\ldots},x_k \right)\right] \approx \int {\ldots} \int \int {\ldots} \int L\left( y_1\right) {\ldots} L\left( y_k\right) \cdot K \left( {\rm z}_1 \right) {\ldots} K \left( {\rm z}_{k-1} \right) F_{{\rm s}_1,{\rm s}_2+h_1{\rm z}_{k-1},{\ldots}, {\rm s}_k+h_1{\rm z}_{k-1}+ {\cdots} +h_{k-1}{\rm z}_1} \left( x_1 -hy_1, {\ldots}, x_k -hy_k\right) d{\rm z}_1 {\ldots} d{\rm z}_{k-1} dy_1 {\ldots} dy_k= F_{{\rm s}_1,{\rm s}_2,{\ldots}, {\rm s}_k} \left( x_1,{\ldots},x_k \right) + O \left( h_{1}^{2}+{\cdots}+ h_{k-1}^{2}+h^2\right)$$

Finally, the approximation of the variance of the continuous estimator will be addressed as given below:

$$ {\rm Var} \left[\tilde{F}_{{\rm s}_1,{\ldots}, {\rm s}_k} \left( x_1,{\ldots},x_k \right)\right] \approx W_1+W_2 $$

with:

$$ W_1 = \sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_k=1}^{n} {\rm E} \left[ A_{i_1,i_2}^{2} {\ldots} A_{i_{k-1},i_k}^{2} \left( {\cal L} \left( \frac{x_1 -X ( {\rm t}_{i_1} )}{h} \right) {\ldots} {\cal L} \left( \frac{x_k -X ( {\rm t}_{i_k} )}{h} \right) - F_{{\rm s}_1, {\ldots},{\rm s}_k } \left( x_1,{\ldots},x_k \right)^2 \right) \right] W_2 = \sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_k=1}^{n} \sum\limits_{j_1=1}^{n} {\ldots} \sum\limits_{j_k=1}^{n} {\rm E} \left[ A_{i_1,i_2} {\ldots} A_{i_{k-1},i_k} A_{j_1,j_2} {\ldots} A_{j_{k-1},j_k}\cdot \left( {\cal L} \left( \frac{x_1 -X ( {\rm t}_{i_1} )}{h} \right) {\ldots} {\cal L} \left( \frac{x_k -X ( {\rm t}_{i_k} )}{h}\right) {\cal L} \left( \frac{x_1 -X ( {\rm t}_{j_1} )}{h} \right) {\ldots} {\cal L} \left( \frac{x_k -X ( {\rm t}_{j_k} )}{h} \right)-F_{{\rm s}_1, {\ldots},{\rm s}_k } \left( x_1,{\ldots},x_k \right)^2 \right) \right] $$

Now, we can combine the arguments used for the bias of the continuous estimator with those applied for the variance of the discrete estimator to check that both terms, W 1 and W 2, are asymptotically negligible, as established next:

$$ W_1 \approx \sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_k=1}^{n} {\rm E} \left[ A_{i_1,i_2}^2 {\ldots} A_{i_{k-1},i_{k}}^2 \int L( y_1) {\ldots} L( y_k)\cdot ( F_{{\rm s}_1, {\ldots},{\rm s}_k} ( x_1 -hy_1 , {\ldots}, x_k -hy_k ) - F_{{\rm s}_1, {\ldots},{\rm s}_k } ( x_1,{\ldots},x_k )^2 ) dy_1 {\ldots} dy_k\right] \approx \sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_k=1}^{n} {\rm E} \left[ A_{i_1,i_2}^2 {\ldots} A_{i_{k-1},i_{k}}^2 ( F_{{\rm s}_1, {\ldots},{\rm s}_k } ( x_1,{\ldots},x_k ) - F_{{\rm s}_1, {\ldots},{\rm s}_k } ( x_1,{\ldots},x_k )^2)\right] \approx\frac{n^{-k} \lambda^{d(k-1)} h_{1}^{-d} {\ldots} h_{k-1}^{-d} ( x_1,{\ldots},x_k ) ( F_{{\rm s}_1,{\ldots}, {\rm s}_k} ( x_1,{\ldots},x_k )- F_{{\rm s}_1,{\ldots}, {\rm s}_k} ( x_1,{\ldots},x_k )^2 )}{ g_0(0)^{k}\left( \int K ( {\rm z} )^2 d{\rm z} \right)^{k-1} }= O \left( n^{-k} \lambda^{d(k-1)} h_{1}^{-d} {\ldots} h_{k-1}^{-d} \right) W_2 \approx \sum\limits_{i_1=1}^{n} {\ldots} \sum\limits_{i_k=1}^{n} \sum\limits_{j_1=1}^{n} {\ldots} \sum\limits_{j_k=1}^{n} {\rm E} \left[ A_{i_1,i_2} {\ldots} A_{i_{k-1},i_{k}} A_{j_1,j_2} {\ldots} A_{j_{k-1},j_{k}}\right]\cdot \int \int {\ldots} \int \int {\ldots} \int L( y_1) {\ldots} L( y_k) L( w_1) {\ldots} L( w_k) ( F_{{\rm s}_1, {\ldots},{\rm s}_k,{\rm s}_1+{\rm t}, {\ldots},{\rm s}_k+{\rm t}} ( x_1 -hy_1 , {\ldots}, x_k -hy_k,x_1 -hw_1 , {\ldots}, x_k -hw_k) - F_{{\rm s}_1, {\ldots},{\rm s}_k } ( x_1,{\ldots},x_k )^2) d{\rm t} dy_1 {\ldots} dy_k \cdot dw_1 {\ldots} dw_k\approx \lambda^{-d} \int ( F_{{\rm s}_1,{\ldots}, {\rm s}_k,{\rm s}_1+{\rm t},{\ldots}, {\rm s}_k+{\rm t}} ( x_1,{\ldots},x_k ,x_1,{\ldots},x_k) - F_{{\rm s}_1,{\ldots}, {\rm s}_k} ( x_1,{\ldots},x_k )^2) d{\rm t}= O (\lambda^{-d})$$

Then, consistency yields for \(\tilde{F}_{{\rm s}_1,{\ldots}, {\rm s}_k} \left( x_1,{\ldots},x_k \right), \) since its bias and variance tend to zero, as the sample size increases.

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García-Soidán, P., Menezes, R. & Rubiños, Ó. Bootstrap approaches for spatial data. Stoch Environ Res Risk Assess 28, 1207–1219 (2014). https://doi.org/10.1007/s00477-013-0808-9

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