# New Robust Cross-Variogram Estimators and Approximations of Their Distributions Based on Saddlepoint Techniques

## Abstract

**:**

## 1. Introduction and Notation

## 2. Preliminary Transformation

#### 2.1. Correlation between ${W}_{t}^{i}$ and ${W}_{s}^{j}$

**a**will be

**t**or

**t**+

**h**and

**b**will be

**s**or

**s**+

**h**, and thus, ${\mu}_{i}=E\left[{Z}_{i}(\mathbf{t}+\mathbf{h})\right]=E\left[{Z}_{i}(\mathbf{t})\right]$ and ${\mu}_{j}=E\left[{Z}_{j}(\mathbf{s}+\mathbf{h})\right]=E\left[{Z}_{j}(\mathbf{s})\right]$, where the equality between the expectations is obtained because of the intrinsic stationary property of the components of

**Z**.

**h**, ${\sigma}_{i}^{2}=V\left[{Z}_{i}(\mathbf{t}+\mathbf{h})\right]=V\left[{Z}_{i}(\mathbf{t})\right]$, and ${\sigma}_{j}^{2}=V\left[{Z}_{j}(\mathbf{s}+\mathbf{h})\right]=V\left[{Z}_{j}(\mathbf{s})\right]$.

**t**+

**h**=

**s**.

**Remark**

**1.**

#### 2.2. Independence of the Observations ${X}_{s}$

#### 2.3. Distribution of the Transformed Variables

**Proposition**

**1.**

**Proposition**

**2.**

## 3. Cross-Variogram $\mathit{M}$-Estimators

#### 3.1. Von Mises Approximation for their Distributions

#### 3.2. Saddlepoint Approximation of the TAIF

## 4. Sample Distribution of the Method-of-Moments Estimator

#### 4.1. Performance of the Theoretical Results with Simulations

#### 4.2. Robustness of the Method-of-Moments-Estimator

**Remark**

**2.**

**h**, that is fixed in advance. If

**h**is small, the number of lags will be large and ${N}_{h}$ will be small. The VOM+SAD approximations obtained in the paper are very accurate, even in this case.

**h**is large, the number of lags will be small and the sample size ${N}_{h}$ will be large. In this case, it is easier to compute the leading term as

## 5. $\mathit{\alpha}$-Trimmed Cross-Variogram Estimator

## 6. Huber’s Cross-Variogram Estimator

`huber`function of the MASS library, [23].

**Example**

**1.**

- (A)
- No contamination, ${Z}_{1}\equiv N(0,1)$;
- (B)
- ${Z}_{1}\equiv 0.95\xb7N(0,1)+0.05\xb7N(0,{5}^{2})$;
- (C)
- ${Z}_{1}\equiv 0.90\xb7N(0,1)+0.10\xb7N(0,{5}^{2})$;
- (D)
- ${Z}_{1}\equiv 0.80\xb7N(0,1)+0.20\xb7N(0,{5}^{2})$;
- (E)
- ${Z}_{1}\equiv 0.95\xb7N(0,1)+0.05\xb7N(0,{20}^{2})$;
- (F)
- ${Z}_{1}\equiv 0.90\xb7N(0,1)+0.10\xb7N(0,{20}^{2})$;
- (G)
- ${Z}_{1}\equiv 0.80\xb7N(0,1)+0.20\xb7N(0,{20}^{2})$.

## 7. Linearized Version of the Cross-Variogram Model

**Example**

**2.**

`prediction.dat`in the R library,

`gstat`.

`ln(Pb)`(natural logarithm of Lead) and

`Ni`(Nickel).

**Example**

**3.**

_{2}), suspended particles with a size less than 10 microns (PM10), and ozone (O

_{3}). These data are obtained from 22 monitoring stations [24,25,26].

`NO`and

`NO`.

_{2}## 8. Conclusions

**h**is small.

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of Proposition**

**1.**

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**Figure 1.**Approximate tail probabilities (in black) and simulated (in red) for the method-of-moments estimator $2\widehat{{\gamma}_{ij}}(\mathbf{h})$ with sample size ${N}_{h}=3$, with no contamination, and with three different degrees of contamination $\u03f5$.

**Figure 2.**Tail distribution of the method-of-moments estimator $2\widehat{{\gamma}_{ij}}(\mathbf{h})$ with sample size ${N}_{h}=3$ and two underlying models: $(1-\u03f5)N(0,1)+\u03f5N(0,{1.1}^{2})$ and $(1-\u03f5)N(0,1)+\u03f5N(0,{1.2}^{2})$, for three different degrees of contamination $\u03f5$.

**Figure 3.**Tail probabilities of the classical method-of-moments cross-variogram estimator $2\widehat{{\gamma}_{ij}}(\mathbf{h})$ (top row of figures) and $0.2$-trimmed cross-variogram estimator $2{\widehat{{\gamma}_{ij}}}_{\alpha}(\mathbf{h})$ (bottom row of figures), with no contamination, $\u03f5=0$, and contaminations $\u03f5=0.15$ and $\u03f5=0.3$.

**Figure 4.**Variogram estimations of Example 1: classical (black), 0.1-trimmed (green) and Huber’s (red), and the variogram model with no contamination.

**Figure 5.**Variogram estimations of Example 1: classical (black), 0.1-trimmed (green) and Huber’s (red), and the variogram model with no contamination.

**Figure 7.**Classical (black) and robust (green and red) variogram estimations for the logarithm of lead and their linearized variograms of Example 2.

**Figure 8.**Classical (black) and robust (green and red) variogram estimations for nickel and their linearized variograms of Example 2.

**Figure 9.**Classical (black) and robust (green and red) cross-variogram estimations, linearized versions, and the classical model (blue) of Example 2.

**Figure 10.**Variogram-crossvariogram matrix of the classical variogram and cross-variogram estimations with the classical model of Example 3.

**Figure 11.**Classical (black) and robust (green and red) cross-variogram estimations of Example 3, with the linearized cross-variogram models.

**Table 1.**Tail probabilities of the VOM+SAD approximation and the exact (simulated) values for the method-of-moments estimator of Figure 1.

No Contamination $\mathit{\u03f5}=0$ | $\mathit{\u03f5}=0.05$ | $\mathit{\u03f5}=0.2$ | ||||
---|---|---|---|---|---|---|

Approximation | Exact | Approximation | Exact | Approximation | Exact | |

$t=0.4$ | 0.26627 | 0.27903 | 0.27101 | 0.28199 | 0.28524 | 0.29635 |

$t=0.6$ | 0.11944 | 0.12512 | 0.12319 | 0.12904 | 0.13443 | 0.13795 |

$t=0.8$ | 0.05052 | 0.05665 | 0.05302 | 0.05895 | 0.06053 | 0.06312 |

$t=0.9$ | 0.03189 | 0.03671 | 0.03386 | 0.03727 | 0.03978 | 0.04304 |

$t=1.0$ | 0.01950 | 0.02451 | 0.02102 | 0.02449 | 0.02560 | 0.02811 |

No Contamination $\mathit{\u03f5}=0$ | $\mathit{\u03f5}=0.05$ | $\mathit{\u03f5}=0.2$ | |
---|---|---|---|

$t=0.4$ | 1.7698 | 1.5292 | 1.5789 |

$t=0.6$ | 0.6492 | 0.6717 | 0.4083 |

$t=0.8$ | 0.6498 | 0.6301 | 0.2764 |

$t=0.9$ | 0.5004 | 0.3542 | 0.3407 |

$t=1.0$ | 0.5136 | 0.3557 | 0.2583 |

**Table 3.**Values of ${S}_{n}={sup}_{\mathbf{h}}\u22252\widehat{\gamma}(\mathbf{h})-2\gamma (\mathbf{h})\u2225$ and its p-value considering a scale contaminated normal with $\u03f5=0.01$ and $g=1.1$ of Example 2.

Log Lead | Nickel | Cross-Variogram | ||||
---|---|---|---|---|---|---|

${\mathit{S}}_{\mathit{n}}$ | p-Value | ${\mathit{S}}_{\mathit{n}}$ | p-Value | ${\mathit{S}}_{\mathit{n}}$ | p-Value | |

Classical | 0.0704076 | 0.052087 | 27.8255 | 0.112065 | 1.160842 | 0.9775347 |

0.1-trimmed mean | 0.0312044 | 1 | 58.3908 | 1 | 0.878346 | 0.7257757 |

Huber | 0.0634437 | 1 | 29.0930 | 1 | 0.894193 | 1 |

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**MDPI and ACS Style**

García-Pérez, A.
New Robust Cross-Variogram Estimators and Approximations of Their Distributions Based on Saddlepoint Techniques. *Mathematics* **2021**, *9*, 762.
https://doi.org/10.3390/math9070762

**AMA Style**

García-Pérez A.
New Robust Cross-Variogram Estimators and Approximations of Their Distributions Based on Saddlepoint Techniques. *Mathematics*. 2021; 9(7):762.
https://doi.org/10.3390/math9070762

**Chicago/Turabian Style**

García-Pérez, Alfonso.
2021. "New Robust Cross-Variogram Estimators and Approximations of Their Distributions Based on Saddlepoint Techniques" *Mathematics* 9, no. 7: 762.
https://doi.org/10.3390/math9070762