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

Algorithm 1

Mathematica source code for computing intermediate correlations for specified 𝐿-correlations. The example is for distributions 𝑗=1 and 𝑘=2 (𝜂₁₂) in Table 1. See Tables 2 and 4.

(* Intermediate Correlation for Distributions 1 and 2. See Table 4. *)

𝜌_𝑗𝑘=0.714645;

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 𝐿-moment Parameters for Distribution 𝑗=1 in Table 1*)

𝛾₁=−0.8429;

𝛾₂=4.6222;

𝛾₃=1.5;

𝛾₄=0.25;

(* Quantile function 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}}]

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

𝜂_𝑗𝑘=2√𝜋∗NIntegrate[𝑋_𝑗∗𝐹_𝑘∗𝑓_𝑗𝑘,{𝑍_𝑗,−10,10},{𝑍_𝑘,−10,10},Method→MultiDimensional]

0.70