A Method for Simulating Burr Type III and Type XII Distributions through 𝐿-Moments and 𝐿-Correlations

Algorithm 1

Mathematica source code for computing intermediate correlations for specified 𝐿-correlations. The example is for distribution 𝑗=1 toward distribution 𝑘=2 (𝜂₁₂). See dashed curves of Distribution 1 and Distribution 2 in Figure 2 (Panel b), specified correlation in Table 8, and intermediate correlation in Table 10.

(* Intermediate Correlation *)

𝑟₁₂= 0.784047;

Needs[“MultivariateStatistics̀ ”]

𝑓₁₂= PDF[MultinormalDistribution[{0, 0}, {{1, 𝑟₁₂}, {𝑟₁₂, 1}}], {𝑍₁, 𝑍₂}];

Φ₁= CDF[NormalDistribution[0, 1], 𝑍₁];

Φ₂= CDF[NormalDistribution[0, 1], 𝑍₂];

(* Parameters for dashed curves of Distribution 1 and Distribution 2 in Figure 2 (Panel b) *)

𝑐₁ = 1.227122;

𝑘₁ = 3.417227;

𝜆₁=0.441134;

𝛿₁=2.543505;

𝑐₂ = −2.944030;

𝑘₂ = 0.126960;

(* Quantile function from (3) *)

𝑦₁=1−Φ₁^(−1/𝑘1)−1^(1/𝑐1);

(* Standardizing constants 𝜆₁ and 𝛿₁ were obtained, respectively, from (19) and (33) *)

𝑥₁=𝛿₁∗(𝑦₁−𝜆₁);

(* Compute the specified 𝐿-correlation *)

𝜂₁₂=2√𝜋∗NIntegrate[𝑥₁∗Φ₂∗𝑓₁₂, {𝑍₁, −8, 8}, {𝑍₂, −8, 8}, Method →{“MultiDimensionalRule”,

“Generators”→ 9}]

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