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

Algorithm 2

Mathematica source code for computing intermediate correlations for specified conventional product moment-based (Pearson) correlations. The example is for distribution 𝑗=1 and distribution 𝑘=2 (𝜌₁₂). See dashed curves of Distribution 1 and Distribution 2 in Figure 2 (Panel a), specified correlation in Table 8 and intermediate correlation in Table 9.

(* Intermediate Correlation *)

𝑟₁₂= 0.868005;

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 a) *)

𝑐₁ = 1.097740;

𝑘₁ = 4.481719;

𝑐₂ = −5.542062;

𝑘₂ = 0.027237;

(* Quantile function from (3) and (2) *)

𝑦₁=1−Φ₁^(−1/𝑘1)−1^(1/𝑐1)

𝑦₂=Φ₂^(−1/𝑘2)−1^(1/𝑐2);

(* Standardizing constants 𝜇₁, 𝜇₂ and 𝜎₁, 𝜎₂ are obtained, respectively, using (14) and (15)

from Headrick et al. [2, page 2211] *)

𝑥₁=(𝑦₁−𝜇₁)/𝜎₁;

𝑥₂=(𝑦₂−𝜇₂)/𝜎₂;

(* Compute the specified conventional product moment-based (Pearson) correlation *)

𝜌₁₂= NIntegrate[𝑥₁∗𝑥₂∗𝑓₁₂, {𝑍₁, −8, 8}, {𝑍₂, −8, 8}, Method →{“MultiDimensionalRule”,

“Generators”→ 9}]

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