A Method for Simulating Nonnormal Distributions with Specified L-Skew, L-Kurtosis, and L-Correlation

Algorithm 2

Mathematica source code for computing intermediate correlations for specified conventional Pearson correlations. The example is for distributions 𝑗=1 and 𝑘=4 (𝜌^∗₃₄) in Figure 4 (Panels 4(a) and 4(d)). See also Table 2(b).

(*Intermediate Correlation*)

𝜌_𝑗𝑘=0.673333;

Needs[“MultivariateStatistics’”]

𝑓_𝑗𝑘=𝑃DF[MultinormalDistribution[{0,0},{{1,𝜌_𝑗𝑘},{𝜌_𝑗𝑘,1}}],{𝑍_𝑗,𝑍_𝑘}];

𝐹_𝑗=𝐶DF[NormalDistribution[0,1],𝑍_𝑗];

𝐹_𝑘=𝐶DF[NormalDistribution[0,1],𝑍_𝑘];

(*Parameters for Distributions 4(a) and 4(d) in Figure 4∗)

𝜅_ℒ𝑗=−0.044817709;

𝜅_ℛ𝑗=0.1704343967;

𝜅_ℒ𝑘=0.0;

𝜅_ℛ𝑘=0.0;

𝑋_𝑗=𝐿og[𝐹_𝑗/(1−𝐹_𝑗)];

𝑋_𝑘=𝐿og[𝐹_𝑘/(1−𝐹_𝑘)];

𝑞_ℒ𝑗=𝑋_𝑗∗Exp[𝜅_ℒ𝑗∗Abs[𝑋_𝑗]];

𝑞_ℛ𝑗=𝑋_𝑗∗Exp[𝜅_ℛ𝑗∗Abs[𝑋_𝑗]];

𝑞_ℒ𝑘=𝑋_𝑘∗Exp[𝜅_ℒ𝑘∗Abs[𝑋_𝑘]];

𝑞_ℛ𝑘=𝑋_𝑘∗Exp[𝜅_ℛ𝑘∗Abs[𝑋_𝑘]];

(*Standardizing constants 𝜆₁ from (3.13) and 𝛼₂ from (A.2) from the appendix*)

𝑆𝑞_ℒ𝑗=(𝑞_ℒ𝑗−𝜆_1𝑗)/𝛼_2𝑗;

𝑆𝑞_ℛ𝑗=(𝑞_ℛ𝑗−𝜆_1𝑗)/𝛼_2𝑗;

𝑆𝑞_ℒ𝑘=(𝑞_ℒ𝑘−𝜆_1𝑘)/𝛼_2𝑘;

𝑆𝑞_ℛ𝑘=(𝑞_ℛ𝑘−𝜆_1𝑘)/𝛼_2𝑘;

(*Compute the specified conventional Pearson correlation*)

𝜌^∗_𝑗𝑘= NIntegrate[Piecewise[{{𝑆𝑞_ℒ𝑗,𝑋_𝑗≤0},{𝑆𝑞_ℛ𝑗,𝑋_𝑗>0}}]∗ Piecewise[{{𝑆𝑞_ℒ𝑘,𝑋_𝑘≤0},

{𝑆𝑞_ℛ𝑘,𝑋_𝑘>0}}]*

𝑓_𝑗𝑘, {𝑍_𝑗,−10,10}, {𝑍_𝑘,−10,10}, Method→MultiDimensional]

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