Multivariate doubly truncated moments for generalized skew-elliptical distributions with applications

Figures & data

Figure 1. $\mathrm{MDTEs}$ of $\mathbf{X}$, $\mathbf{Y}$, $\mathbf{Z}$ and $\mathbf{S}$ for $(\mathit{a}\mathbf{,}\mathit{b})$.

Figure 2. The trace plots of the Monte Carlo estimates of the $2$ elements of μ${ }^{\ast }$ (${\mu }_1^{\ast }$ (left) and ${\mu }_2^{\ast }$ (right)).

Figure 3. The trace plots of the Monte Carlo estimates of the three elements of Σ ${}^{\ast }$ (${\sigma }_{11}^{\ast }$ (left), ${\sigma }_{22}^{\ast }$ (center) and ${\sigma }_{12}^{\ast }$ (right)).

Figure 4. The trace plots of the Monte Carlo estimates of the $2$ elements of μ ${ }^{†}$ (${\mu }_1^{†}$ (left) and ${\mu }_2^{†}$ (right)).

Figure 5. The trace plots of the Monte Carlo estimates of the three elements of ${\Sigma }^{†}$ (${\sigma }_{11}^{†}$ (left), ${\sigma }_{22}^{†}$ (center) and ${\sigma }_{12}^{†}$ (right)).

Figure 6. The trace plots of the Monte Carlo estimates of the $2$ elements of μ${ }^{\circ }$ (${\mu }_1^{\circ }$ (left) and ${\mu }_2^{\circ }$ (right)).

Figure 7. The trace plots of the Monte Carlo estimates of the three elements of Σ${ }^{°}$ (${\sigma }_{11}^{\circ }$ (left), ${\sigma }_{22}^{\circ }$ (center) and ${\sigma }_{12}^{\circ }$ (right)).

Figure 8. The trace plots of the Monte Carlo estimates of the $2$ elements of μ ${ }^{‡}$ (${\mu }_1^{‡}$ (left) and ${\mu }_2^{‡}$ (right)).

Figure 9. The trace plots of the Monte Carlo estimates of the three elements of Σ${\text{math}}^{\mathit{‡}}$ (${\sigma }_{11}^{‡}$ (left), ${\sigma }_{22}^{‡}$ (center) and ${\sigma }_{12}^{‡}$ (right)).

Table 1. Fitted distributions and their information criterion values to the logarithm of adjusted prices of different sectors.

Subclass	SMN$|$		SMSN and SSMN
Sample	$\mathbf{U}$	$\mathbf{W}$	$\mathbf{U}$	$\mathbf{W}$
Model	Normal	Normal	Skew-Normal	Skew-Normal
AIC	−1166.7755	−1500.5947	−1255.5680	−1557.5896
BIC	−1135.0106	−1468.8299	−1213.2150	−1515.2365

Figure 10. PP plot for the skew-normal distribution for the logarithm of adjusted prices of companies in the financials (left) and energy sectors (right).

Figure 11. Bivariate scatter plots with contour lines for the skew-normal distribution for the logarithm of adjusted prices of companies in the financials (left) and energy sectors (right).

Table 2. The VaRs of $\mathbf{U}$ and $\mathbf{W}$ for $\mathit{q}=(0.95,0.95,0.95{)}^{\mathrm{T}}$.

$\mathit{i}$	$1$	$2$	$3$
$\mathit{Va}{R}_{\mathit{q}}(\mathbf{U}{)}_i$	5.054407	4.509904	4.762440
$\mathit{Va}{R}_{\mathit{q}}(\mathbf{W}{)}_i$	4.240676	3.476439	4.887053

Table 3. The MTCEs of $\mathbf{U}$ and $\mathbf{W}$ for $\mathit{q}=(0.95,0.95,0.95{)}^{\mathrm{T}}$.

$\mathit{i}$	$1$	$2$	$3$
$\mathrm{MTC}{\mathrm{E}}_{\mathit{q}}(\mathbf{U}{)}_i$	5.389089	4.647451	4.930082
$\mathrm{MTC}{\mathrm{E}}_{\mathit{q}}(\mathbf{W}{)}_i$	4.469574	3.845761	5.047621