An L-Moment-Based Analog for the Schmeiser-Deutsch Class of Distributions

Algorithm 2

Mathematica source code for computing intermediate correlations for specified conventional Pearson correlations. The example is for distributions 𝑗=1 and 𝑘=2 (𝜌^∗₁₂) in Table 1. See also Tables 2 and 3.

(* Intermediate Correlation for Distributions 1 and 2. See Table 3. *)

𝜌_𝑗𝑘= 0.761371;

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];

(* S-D C-Moment Parameters Distributions 𝑗=1 and 𝑘=2 in Table 1*)

𝛾_1𝑗=−0.8427;

𝛾_2𝑗=4.6213;

𝛾_3𝑗=1.5;

𝛾_4𝑗=0.25;

𝛾_1𝑘=0.0;

𝛾_2𝑘=21.166;

𝛾_3𝑘=3.0;

𝛾_4𝑘=0.5;

(* Quantile functions for (1.1) *)

𝑞1_𝑗=𝛾_1𝑗−𝛾_2𝑗∗(𝛾_4𝑗−𝐹_𝑗)^𝛾3𝑗;

𝑞2_𝑗=𝛾_1𝑗+𝛾_2𝑗∗(𝐹_𝑗−𝛾_4𝑗)^𝛾3𝑗;

𝑋_𝑗= Piecewise[{{𝑞1_𝑗,𝐹_𝑗≤𝛾_4𝑗},{𝑞2_𝑗,𝐹_𝑗>𝛾_4𝑗}}]

𝑞1_𝑘=𝛾_1𝑘−𝛾_2𝑘∗(𝛾_4𝑘−𝐹_𝑘)^𝛾3𝑘;

𝑞2_𝑘=𝛾_1𝑘+𝛾_2𝑘∗(𝐹_𝑘−𝛾_4𝑘)^𝛾3𝑘;

𝑋_𝑘= Piecewise[{{𝑞1_𝑘,𝐹_𝑘≤𝛾_4𝑘},{𝑞2_𝑘,𝐹_𝑘>𝛾_4𝑘}}]

(* Compute the specified correlation in Table 2 for Distributions 1 and 2. *)

𝜌^∗_𝑗𝑘=NIntegrate[𝑋_𝑗∗𝑋_𝑘∗𝑓_𝑗𝑘, {𝑍_𝑗,−10,10}, {𝑍_𝑘,−10,10}, Method→MultiDimensional]

0.70