Non-normal Data Simulation using Piecewise Linear Transforms

Figures & data

Figure 1. Graph of the continuous piecewise linear function $H_1(x)$

Figure 2. The density of $Y=H_1(Z)$

Figure 3. Graph of the piecewise linear function $H_2(x)$.

Figure 4. The density of $Y=H_2(Z)$.

Figure 5. Graph of the piecewise linear function $H_3(x)$.

Figure 6. The density of $Y=H_3(Z)$.

Figure 7. The correlation among piecewise linear transforms of bivariate normal variables with correlation $\rho$.

Figure 8. Graph of the density of $Z$ and the standardized version of $Z^3$ when $Z\sim N(0,1)$.

Table 1. Correlation matrix and marginal skewnesses and excess kurtosi for the attitude dataset

	Rating	Complaints	Privileges	Learning	Raises	Critical	Advance
Rating	$1$	$.825$	$.426$	$.624$	$.590$	$.156$	$.155$
complaints		$1$	$.558$	$.597$	$.669$	$.188$	$.225$
privileges			$1$	$.493$	$.445$	$.147$	$.343$
learning				$1$	$.640$	$.116$	$.532$
raises					$1$	$.377$	$.574$
critical						$1$	$.283$
advance							$1$
skewness	−0.358	−0.215	0.379	−0.054	0.198	−0.866	0.850
kurtosis	−0.766	−0.677	−0.411	−1.223	−0.599	0.166	0.466

Figure 9. Plots for $n=100$ dataset simulated from PLSIM.

Figure 10. Histograms of sample estimates of univariate kurtosis based on 1000 simulated datasets of size $n=30$. The red line represents the population kurtosis derived from the attitude dataset.

Figure 11. Histogram of multivariate kurtosis $b_{2,p}$ based on 1000 simulated datasets of size $n=30$. The red line represents $b_{2,p}$ calculated for the attitude dataset.