International Scholarly Research Notices

Research Article

A Doubling Method for the Generalized Lambda Distribution

Algorithm 2

Mathematica source code for computing intermediate correlations for specified conventional Pearson correlations. The example is for distributions 𝑗 = 3 and π‘˜ = 4 ( 𝜌 βˆ— 3 4 ) in Figure 4. See also Table 2(b).
Ξ¦ 𝑗 = C DF[NormalDistribution [ 0,1], 𝑍 𝑗 ];
Ξ¦ π‘˜ = C DF[NormalDistribution [ 0,1], 𝑍 π‘˜ ];
πœ† β„’ 𝑗 = 0 . 2 7 7 9 7 3 3 5 9 8 2 8 3 2 3 2 ;
πœ† β„› 𝑗 = βˆ’ 0 . 0 8 3 2 9 4 7 9 1 4 3 3 1 2 7 6 2 ;
πœ† β„’ π‘˜ = 0 . 3 6 6 2 5 9 9 0 6 9 2 9 8 9 9 4 ;
πœ† β„› π‘˜ = 0 . 1 2 9 7 7 7 1 2 6 3 0 8 0 0 0 4 ;
π‘ž β„’ 𝑗 = ( Ξ¦ πœ† β„’ 𝑗 𝑗 βˆ’ ( 1 βˆ’ Ξ¦ 𝑗 ) πœ† β„’ 𝑗 ) / ( πœ† β„’ 𝑗 Γ— 2 ( 3 / 2 ) βˆ’ πœ† β„’ 𝑗 / √ πœ‹ ) ;
π‘ž β„› 𝑗 = ( Ξ¦ πœ† β„› 𝑗 𝑗 βˆ’ ( 1 βˆ’ Ξ¦ 𝑗 ) πœ† β„› 𝑗 ) / ( πœ† β„› 𝑗 Γ— 2 ( 3 / 2 ) βˆ’ πœ† β„› 𝑗 / √ πœ‹ ) ;
π‘ž β„’ π‘˜ = ( Ξ¦ πœ† β„’ π‘˜ π‘˜ βˆ’ ( 1 βˆ’ Ξ¦ π‘˜ ) πœ† β„’ π‘˜ ) / ( πœ† β„’ π‘˜ Γ— 2 ( 3 / 2 ) βˆ’ πœ† β„’ π‘˜ / √ πœ‹ ) ;
π‘ž β„› π‘˜ = ( Ξ¦ πœ† β„› π‘˜ π‘˜ βˆ’ ( 1 βˆ’ Ξ¦ π‘˜ ) πœ† β„› π‘˜ ) / ( πœ† β„› π‘˜ Γ— 2 ( 3 / 2 ) βˆ’ πœ† β„› π‘˜ / √ πœ‹ ) ;
(* Standardizing constants 𝛼 1 and 𝛼 2 from Equation ( A . 2 ) in the Appendix *)
𝑋 β„’ 𝑗 = ( π‘ž β„’ 𝑗 βˆ’ 𝛼 1 𝑗 ) / 𝛼 2 𝑗 ;
𝑋 β„› 𝑗 = ( π‘ž β„› 𝑗 βˆ’ 𝛼 1 𝑗 ) / 𝛼 2 𝑗 ;
𝑋 β„’ π‘˜ = ( π‘ž β„’ π‘˜ βˆ’ 𝛼 1 π‘˜ ) / 𝛼 2 π‘˜ ;
𝑋 β„› π‘˜ = ( π‘ž β„› π‘˜ βˆ’ 𝛼 1 π‘˜ ) / 𝛼 2 π‘˜ ;
(* Intermediate Correlation *)
𝜌 𝑗 π‘˜ = 0 . 4 0 6 7 8 6 ;
Needs[β€œMultivariateStatistics`”]
𝑓 𝑗 π‘˜ = P DF [ MultinormalDistribution [ { 0,0}, { { 1, 𝜌 𝑗 π‘˜ } , { 𝜌 𝑗 π‘˜ ,1 } } ] , { 𝑍 𝑗 , 𝑍 π‘˜ } ] ;
(* Compute the specified conventional Pearson correlation *)
𝜌 βˆ— 𝑗 π‘˜ = NIntegrate [ Piecewise [ { { 𝑋 β„’ 𝑗 , Ξ¦ 𝑗 ≀ 0 . 5 } , { 𝑋 β„› 𝑗 , Ξ¦ 𝑗 > 0 . 5 } } ] Γ— Piecewise [ { { 𝑋 β„’ π‘˜ , Ξ¦ π‘˜ ≀
0 . 5 } , { 𝑋 β„› π‘˜ , Ξ¦ π‘˜ > 0 . 5 } } ] Γ— 𝑓 𝑗 π‘˜ , { 𝑍 𝑗 , βˆ’10, 10}, { 𝑍 π‘˜ , βˆ’10, 10}, Method β†’ MultiDimensional ]
0.40