# Birnbaum-Saunders Quantile Regression Models with Application to Spatial Data

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## Abstract

**:**

## 1. Introduction

## 2. Quantile Regression

## 3. The Univariate Birnbaum-Saunders Distribution

- (i)
- $\mathrm{E}\left[T\right]=\varrho (1+{\alpha}^{2}/2)$.
- (ii)
- $\mathrm{Var}\left[T\right]={\varrho}^{2}{\alpha}^{2}(1+5{\alpha}^{2}/4)$.
- (iii)
- $bT\sim \mathrm{BS}(\alpha ,b\varrho )$, for $b>0$.
- (iv)
- $1/T\sim \mathrm{BS}(\alpha ,1/\varrho )$.
- (v)
- $W={Z}^{2}=(1/{\alpha}^{2})(T/\varrho +\varrho /T-2)\sim {\chi}^{2}(1)$, with $\mathrm{E}\left[W\right]=1$ and $\mathrm{Var}\left[W\right]=2$.

## 4. The Multivariate BS Distribution and a New Parametrization

**Theorem**

**1.**

- (i)
- ${T}_{i}\sim \mathrm{BS}({\alpha}_{i},{Q}_{i})$, for $i=\overline{1,n}$.
- (ii)
- $({T}_{i},{T}_{j})\sim {\mathrm{BS}}_{2}({\mathit{\alpha}}^{(i,j)},{\mathit{Q}}^{(i,j)},{\mathbf{\Sigma}}^{(i,j)})$, where ${\mathit{\alpha}}^{(i,j)}=({\alpha}_{i},{\alpha}_{j})$, ${\mathit{Q}}^{(i,j)}=({Q}_{i},{Q}_{j})$ and ${\mathbf{\Sigma}}^{(i,j)}$ is a $2\times 2$ matrix with ones in its diagonal and its other elements equal to element $(i,j)$ of the matrix Σ.
- (iii)
- $$\mathrm{Cov}[{T}_{i},{T}_{j}]={\displaystyle \frac{4{\alpha}_{i}{\alpha}_{j}{Q}_{i}{Q}_{j}}{{\gamma}_{{\alpha}_{i}}^{2}{\gamma}_{{\alpha}_{j}}^{2}}}\left[{\alpha}_{i}{\alpha}_{j}{\sigma}_{ij}^{2}+4I({\alpha}_{i},{\alpha}_{j},{\sigma}_{ij})\right],\phantom{\rule{1.em}{0ex}}i,j=\overline{1,n},$$
- (iv)
- The variance-covariance matrix of $\mathit{T}$ is $\mathit{Var}\left[\mathit{T}\right]=4\Omega \odot \left(\mathbf{\Sigma}\odot \mathbf{\Sigma}\odot \Xi +4\mathit{U}\right)$, where $\Omega =({\omega}_{ij})$, $\Xi =({\xi}_{ij})$ and $\mathit{U}=({u}_{ij})$ have elements ${\omega}_{ij}={\alpha}_{i}^{2}{\alpha}_{j}^{2}{Q}_{i}{Q}_{j}/({\gamma}_{{\alpha}_{i}}^{2}{\gamma}_{{\alpha}_{j}}^{2})$, ${\xi}_{ij}={\alpha}_{i}{\alpha}_{j}$ and ${u}_{ij}=I({\alpha}_{i},{\alpha}_{j},{\sigma}_{ij})$, respectively, for $i,j=\overline{1,n}$, and ⊙ is the Hadamard product. If ${T}_{1},\dots ,{T}_{n}$ are independent random variables, then $\mathit{Var}\left[\mathit{T}\right]=4\mathbf{D}({\u03f5}_{ii})$, where $\mathbf{D}({\u03f5}_{ii})=\mathit{diag}({\u03f5}_{11},\dots ,{\u03f5}_{nn})$, that is, $\mathbf{D}$ is a diagonal matrix with elements ${\u03f5}_{ii}={\alpha}_{i}^{2}{Q}_{i}^{2}({\alpha}_{i}^{2}+4I({\alpha}_{i},{\alpha}_{i},1))/{\gamma}_{{\alpha}_{i}}^{4}$.

**Proof.**

**Corollary**

**1.**

- (i)
- $$E\left[{T}_{1}{T}_{2}\right]=\frac{4{Q}_{1}{Q}_{2}}{{\gamma}_{{\alpha}_{1}}^{2}{\gamma}_{{\alpha}_{2}}^{2}}\left[4+2({\alpha}_{1}^{2}+{\alpha}_{2}^{2})+{\alpha}_{1}^{2}{\alpha}_{2}^{2}(1+{\sigma}^{2})+4{\alpha}_{1}{\alpha}_{2}\phantom{\rule{0.166667em}{0ex}}I({\alpha}_{1},{\alpha}_{2},\sigma )\right],$$
- (ii)
- $$\mathit{Cov}[{T}_{1},{T}_{2}]=\frac{4{\sigma}^{2}{\alpha}_{1}{\alpha}_{2}{Q}_{1}{Q}_{2}}{{\gamma}_{{\alpha}_{1}}^{2}{\gamma}_{{\alpha}_{2}}^{2}}\left[{\alpha}_{1}{\alpha}_{2}{\sigma}^{2}+4I({\alpha}_{1},{\alpha}_{2},\sigma )\right].$$
- (iii)
- $$\mathit{Corr}({T}_{1},{T}_{2})={\displaystyle \frac{{\alpha}_{1}{\alpha}_{2}{\sigma}^{2}+4I({\alpha}_{1},{\alpha}_{2},\sigma )}{\sqrt{4+5{\alpha}_{1}^{2}}\sqrt{4+5{\alpha}_{2}^{2}}}}.$$

## 5. Formulation of the Spatial Model

## 6. Estimation of Model Parameters

`optim`and

`optimx`implemented in the

`R`software; see www.R-project.org and [42]. The signs of the determinants of the corresponding Hessian matrix and of its minors were also checked to ensure that a valid maximum has been attained.

## 7. Model Checking

## 8. Empirical Illustrative Example

## 9. Conclusions and Future Works

- (i)
- A new parameterization of the multivariate Birnbaum-Saunders distribution has been established.
- (ii)
- A novel Birnbaum–Saunders spatial quantile regression model has been proposed and derived.
- (iii)
- We have developed maximum likelihood estimation for the parameters of the proposed model.
- (iv)
- A randomized quantile residual has been used for model checking. We have utilized the Wilson–Hilferty approximation for our spatial model residuals to evaluate adequacy model.
- (v)
- The obtained results have been applied to a real data set illustrating its potential usages.

- (i)
- A global test for independence might be stated based on ${\mathrm{H}}_{0}:{\sigma}_{ij}=0$ (or $\mathbf{\Sigma}={\mathit{I}}_{n}$, the $n\times n$ identity matrix). Specifically, let ${L}_{\mathrm{full}}$ be the likelihood function for the full model and ${L}_{\mathrm{reduced}}$ be the likelihood function for the reduced model (under ${\mathrm{H}}_{0}$ indicating independence). Subsequently, we can use the likelihood ratio statistic $\Lambda ={L}_{\mathrm{reduced}}/{L}_{\mathrm{full}}$ to test ${\mathrm{H}}_{0}$. Thus, instead of using the asymptotic distribution of $-2log(\Lambda )$, which is unknown, a bootstrap test can be employed.
- (ii)
- In addition, we can consider ${\mathrm{H}}_{0}:\phi =0$ versus ${\mathrm{H}}_{1}:\phi >0$. In this case, the asymptotic distribution of $-2log(\Lambda )$ under ${\mathrm{H}}_{0}$ is an equally weighted mixture of chi-square distributions with zero and one degree of freedom, whose critical value is 2.7055 at a significance level of 5% [47]. In the spatial case, such a distribution might also be unknown, so that the bootstrap technique can be employed.
- (iii)
- it is of interest to study details of the asymptotic behavior and performance of maximum likelihood estimators [48]. However, applicability of asymptotic frameworks to spatial data is not an easy aspect. This is due to there being at least two relevant frameworks, which can behave quite differently when estimating the spatial dependence parameters; see details about these asymptotic frameworks and their implications in [49].
- (iv)
- The Birnbaum–Saunders distribution is based on the normal distribution and then parameter estimation in spatial quantile regression models can be affected by atypical cases. Thus, robust estimation to these cases, for example based on the Birnbaum–Saunders-t distribution, can be considered to decrease their effects; see [50].
- (v)
- Besides fixed effects that are added to the modeling by regression, random effects can also be added by mixed models, which may produce a more sophisticated Birnbaum-Saunders spatial quantile regression model and closer to reality [51].
- (vi)

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Score Vector and Fisher Information Matrix

#### Appendix A.1. Score Vector

#### Appendix A.2. Information Matrix

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**Figure 1.**Plots of the reparametrized bivariate BS PDF for ${\alpha}_{i}=0.5$ (

**a**), ${\alpha}_{i}=0.8$ (

**b**), ${\alpha}_{i}=1.5$ (

**c**) with ${Q}_{i}=1.0$, and ${Q}_{i}=0.5$ (

**d**), ${Q}_{i}=0.8$ (

**e**), ${Q}_{i}=1.5$ (

**f**) with ${\alpha}_{i}=1.0$, for $i=1,2$ and $\sigma =0.9$.

**Figure 2.**Plots of the reparametrized bivariate BS PDF for ${\alpha}_{i}=0.5$ and ${Q}_{i}=1.0$, for $i=1,2$, with $\sigma =0.9$, (

**a**–

**d**) are seen from different angles.

**Figure 3.**Histogram (

**a**), boxplot (

**b**), scatterplot (

**c**), and semi-variogram (

**d**) for the response variable with chemical data.

**Figure 5.**Three-dimensional (

**a**) and two-dimensional (

**b**) scatterplots estimated versus observed response values with chemical data.

Model | Shape Parameter | Correlation Function |
---|---|---|

Exponential | $\delta =0.5$ | $\sigma (h)=exp(-\phi h)$ |

Whittle | $\delta =1.0$ | $\sigma (h)=\phi h{K}_{1}(\phi h)$ |

Gaussian | $\delta \to \infty $ | $\sigma (h)=exp(-{(\phi h)}^{2})$ |

Model | $\mathit{\ell}(\widehat{\mathit{\theta}})$ | CAIC | BIC |
---|---|---|---|

Gaussian | −32.1411 | 70.5900 | 77.5024 |

BS–identity link | −36.3659 | 81.2513 | 90.3587 |

BS–logarithm link | −36.3659 | 81.2513 | 90.3587 |

BS–square root link | −24.9112 | 58.3419 | 67.4493 |

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**MDPI and ACS Style**

Sánchez, L.; Leiva, V.; Galea, M.; Saulo, H. Birnbaum-Saunders Quantile Regression Models with Application to Spatial Data. *Mathematics* **2020**, *8*, 1000.
https://doi.org/10.3390/math8061000

**AMA Style**

Sánchez L, Leiva V, Galea M, Saulo H. Birnbaum-Saunders Quantile Regression Models with Application to Spatial Data. *Mathematics*. 2020; 8(6):1000.
https://doi.org/10.3390/math8061000

**Chicago/Turabian Style**

Sánchez, Luis, Víctor Leiva, Manuel Galea, and Helton Saulo. 2020. "Birnbaum-Saunders Quantile Regression Models with Application to Spatial Data" *Mathematics* 8, no. 6: 1000.
https://doi.org/10.3390/math8061000