Multivariate BirnbaumSaunders Distributions: Modelling and Applications
Abstract
:1. Introduction and Literature Review
2. Univariate Birnbaum–Saunders Distributions
2.1. Genesis and Features of the Univariate BS Distribution
 (A1)
 $k\phantom{\rule{0.166667em}{0ex}}T\sim \mathrm{BS}(\alpha ,k\phantom{\rule{0.166667em}{0ex}}\lambda )$, with $k\in {\mathbb{R}}_{+}$.
 (A2)
 $1/T\sim \mathrm{BS}(\alpha ,1/\lambda )$.
 (A3)
 ${V}^{2}=[T/\lambda +\lambda /T2]/{\alpha}^{2}\sim {\chi}^{2}\left(1\right)$, that is, ${V}^{2}$ follows a ${\chi}^{2}$ distribution with one degree of freedom.
2.2. Univariate LogBS Distribution and BS Modelling
 (B1)
 $Y=\mu +2\phantom{\rule{0.166667em}{0ex}}\mathrm{arcsinh}(\alpha \phantom{\rule{0.166667em}{0ex}}W/2)\sim \mathrm{logBS}(\alpha ,\mu )$, with $W\sim \mathrm{N}(0,1)$, that is, a random variable with logBS distribution can be obtained directly from a random variable with standard normal distribution.
 (B2)
 $W=B(Y;\alpha ,\mu )=[2/\alpha ]\mathrm{sinh}\left(\right[Y\mu ]/2)\sim \mathrm{N}(0,1)$.
 (B3)
 ${W}^{2}={B}^{2}(Y;\alpha ,\mu )\sim {\chi}^{2}\left(1\right)$, that is, ${V}^{2}$ follows a ${\chi}^{2}$ distribution with one degree of freedom.
2.3. Illustration
3. Multivariate Birnbaum–Saunders Distributions
3.1. Multivariate Normal Distribution
3.2. Multivariate BS Distribution
 (C1)
 $k\phantom{\rule{0.166667em}{0ex}}\mathit{T}\sim {\mathrm{BS}}_{m}(\mathit{\alpha},k\phantom{\rule{0.166667em}{0ex}}\mathit{\lambda},\mathbf{\Gamma})$, with $k\in {\mathbb{R}}_{+}$.
 (C2)
 ${\mathit{T}}^{*}={(1/{T}_{1},\dots ,1/{T}_{m})}^{\top}\sim {\mathrm{BS}}_{m}(\mathit{\alpha},{\mathit{\lambda}}^{*},\mathbf{\Gamma})$, with ${\mathit{\lambda}}^{*}={(1/{\lambda}_{1},\dots ,1/{\lambda}_{m})}^{\top}$.
 (C3)
 ${A}^{\top}(\mathit{T};\mathit{\alpha},\mathit{\lambda}){\mathbf{\Gamma}}^{1}A(\mathit{T};\mathit{\alpha},\mathit{\lambda})\sim {\chi}^{2}\left(m\right)$.
3.3. Multivariate LogBS Distribution
 (D1)
 $\mathit{D}\left(\mathit{\alpha}\right)\phantom{\rule{0.166667em}{0ex}}B(\mathit{Y};\mathit{\alpha},\mathit{\mu})\sim {\mathrm{N}}_{m}(\mathbf{0},\mathit{D}\left(\mathit{\alpha}\right)\mathbf{\Gamma}\mathit{D}\left(\mathit{\alpha}\right))$, where $\mathit{D}\left(\mathit{\alpha}\right)=\mathrm{diag}({\alpha}_{1},\dots ,{\alpha}_{m})$ and$$\mathit{D}\left(\mathit{\alpha}\right)\mathbf{\Gamma}\mathit{D}\left(\mathit{\alpha}\right)=\left(\begin{array}{cccc}{\alpha}_{1}^{2}& {\alpha}_{1}{\alpha}_{2}{\rho}_{12}& \cdots & {\alpha}_{1}{\alpha}_{m}{\rho}_{1m}\\ {\alpha}_{1}{\alpha}_{2}{\rho}_{12}& {\alpha}_{2}^{2}& \cdots & {\alpha}_{2}{\alpha}_{m}{\rho}_{2m}\\ \vdots & \vdots & \ddots & \vdots \\ {\alpha}_{1}{\alpha}_{m}{\rho}_{1m}& {\alpha}_{2}{\alpha}_{m}{\rho}_{2m}& \cdots & {\alpha}_{m}^{2}\end{array}\right).$$
 (D2)
 ${B}^{\top}(\mathit{Y};\mathit{\alpha},\mathit{\mu}){\mathbf{\Gamma}}^{1}B(\mathit{Y};\mathit{\alpha},\mathit{\mu})\sim {\chi}^{2}\left(m\right)$, that is, a ${\chi}^{2}$ distribution with m degrees of freedom.
3.4. Mahalanobis Distance and Generation of LogBS Random Vectors
Algorithm 1 Generator of random vectors from multivariate logBS distributions. 

3.5. Illustration
4. Regression Modelling Based on Multivariate Birnbaum–Saunders Distributions
4.1. Formulation
4.2. Illustration
Algorithm 2 PP plots with acceptance bands for testing normality. 

5. Spatial Modelling Based on Multivariate BS Distributions
5.1. Formulation
5.2. Illustration
6. Multivariate Birnbaum–Saunders Control Charts
6.1. Formulation
Algorithm 3 Computation of BS control chart limits in Phase I. 

Algorithm 4 Process monitoring using the multivariate BS chart in Phase II. 

6.2. Illustration
7. Discussion, Conclusions and Future Research
Acknowledgments
Author Contributions
Conflicts of Interest
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Parameter  

$\mathit{\alpha}$  ${\mathit{\beta}}_{\mathbf{11}}$  ${\mathit{\beta}}_{\mathbf{12}}$  ${\mathit{\beta}}_{\mathbf{21}}$  ${\mathit{\beta}}_{\mathbf{22}}$  $\mathit{\rho}$  
ML estimate  0.147407  10.897981  15.524423  −0.005647  −0.005930  0.972392 
Standard error  0.014813  0.236175  0.235865  0.000333  0.000333  0.005219 
pvalue  <0.001  <0.001  <0.001  <0.001  <0.001  <0.001 
Removed case(s)  Matérn Order  $\widehat{\mathit{\alpha}}$  $\widehat{\mathit{\mu}}$  $\widehat{{\mathit{\phi}}_{1}}$  $\widehat{{\mathit{\phi}}_{2}}$  $\widehat{{\mathit{\phi}}_{3}}$  $\widehat{\mathit{a}}$  GA  $\mathit{\kappa}$ 

0.997  2.807  0.134  0.020  177.940  1053.405      
None  $\delta =2.5$  (3.521)  (0.082)  (0.946)  (0.142)  (0.0000014)  (0.0000083)  
[0.389]  [<0.001]  [0.444]  [0.444]  [<0.001]  [<0.001]  
0.993  2.831  0.125  0.016  108.655  643.238  0.90  0.72  
#2  $\delta =2.5$  (4.097)  (0.059)  (1.031)  (0.132)  (0.0000008)  (0.0000047)  
[0.404]  [<0.001]  [0.452]  [0.452]  [<0.001]  [<0.001]  
0.996  2.824  0.125  0.019  152.374  902.054  0.97  0.92  
#48  $\delta =2.5$  (3.417)  (0.073)  (0.856)  (0.133)  (0.0000012)  (0.0000071)  
[0.386]  [<0.001]  [0.442]  [0.443]  [<0.001]  [<0.001]  
0.997  2.817  0.122  0.028  308.828  1235.312  0.84  0.69  
#94  $\delta =1$  (3.417)  (0.073)  (0.856)  (0.133)  (0.0000012)  (0.0009092)  
[0.374]  [<0.001]  [0.436]  [0.437]  [<0.001]  [<0.001]  
0.991  2.845  0.071  0.060  81.052  243.156  0.65  0.31  
#2, #48  $\delta =0.5$ (exponential)  (6.149)  (0.046)  (0.882)  (0.746)  (0.0000884)  (0.0002652)  
[0.436]  [<0.001]  [0.468]  [0.469]  [<0.001]  [<0.001]  
0.995  2.840  0.097  0.038  182.059  546.177  0.72  0.45  
#2, #94  $\delta =0.5$ (exponential)  (3.308)  (0.063)  (0.644)  (0.254)  (0.0002974)  (0.0008922)  
[0.382]  [<0.001]  [0.440]  [0.441]  [<0.001]  [<0.001]  
0.994  2.834  0.114  0.024  144.768  856.737  0.80  0.60  
#48, #94  $\delta =2.5$  (2.980)  (0.077)  (0.683)  (0.145)  (0.0000010)  (0.0000059)  
[0.370]  [<0.001]  [0.434]  [0.435]  [<0.001]  [<0.001]  
0.982  2.855  0.085  0.041  143.699  431.097  0.66  0.35  
#2, #48, #94  $\delta =0.5$ (exponential)  (3.442)  (0.056)  (0.597)  (0.287)  (0.0002051)  (0.0006153)  
[0.388]  [<0.001]  [0.444]  [0.443]  [<0.001]  [<0.001] 
Model  $\mathit{\ell}(\widehat{\mathit{\theta}})$  AIC  BIC  $2\mathrm{log}\left({\mathit{B}}_{12}\right)$ 

BS  $332.576$  $675.152$  $688.276$  28.224 
Gaussian  $349.000$  $706.000$  $716.500$  – 
$2\mathrm{log}\left({\mathit{B}}_{12}\right)$  Evidence in Favour of ${\mathbf{M}}_{1}$ 

<0  Negative (${\mathrm{M}}_{2}$ is accepted) 
$[0,2)$  Weak 
$[2,6)$  Positive 
$[6,10)$  Strong 
≥10  Very strong 
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Aykroyd, R.G.; Leiva, V.; Marchant, C. Multivariate BirnbaumSaunders Distributions: Modelling and Applications. Risks 2018, 6, 21. https://doi.org/10.3390/risks6010021
Aykroyd RG, Leiva V, Marchant C. Multivariate BirnbaumSaunders Distributions: Modelling and Applications. Risks. 2018; 6(1):21. https://doi.org/10.3390/risks6010021
Chicago/Turabian StyleAykroyd, Robert G., Víctor Leiva, and Carolina Marchant. 2018. "Multivariate BirnbaumSaunders Distributions: Modelling and Applications" Risks 6, no. 1: 21. https://doi.org/10.3390/risks6010021