Journal of Probability and Statistics

Research Article

A Characterization of Power Method Transformations through L-Moments

Table 4

Overall average estimates (Est.), bootstrap confidence intervals (C.I.), standard errors (SE), and percentage of relative bias (RBias%) for the conventional moment-based statistics ( 𝑔 ) and L-moment ratios ( 𝑡 ).
(a)
Parameters ( 𝜒 2 𝑑 𝑓 = 8 ) and coefficients
𝛾 1 = 0 𝑐 1 = − 0 . 1 6 3 9 6 8 𝜆 1 = 0 𝑐 1 = − 0 . 1 6 9 1 6 0
𝛾 2 = 1 𝑐 2 = 0 . 9 5 0 7 9 4 𝜆 2 √ = 1 / 𝜋 𝑐 2 = 0 . 9 8 0 8 9 7
𝛾 3 = 1 𝑐 3 = 0 . 1 6 5 3 9 1 𝜏 3 = 0 . 1 6 4 6 6 6 𝑐 3 = 0 . 1 7 0 6 2 7
𝛾 4 = 1 . 5 𝑐 4 = 0 . 0 0 7 3 4 5 𝜏 4 = 0 . 1 3 1 2 3 7 𝑐 4 = 0 . 0 0 7 5 7 7
𝛾 5 = 3 𝑐 5 = − 0 . 0 0 0 4 7 4 𝜏 5 = 0 . 0 5 1 1 9 4 𝑐 5 = − 0 . 0 0 0 4 8 9
𝛾 6 = 7 . 5 𝑐 6 = 0 . 0 0 0 0 1 4 𝜏 6 = 0 . 0 4 8 3 3 4 𝑐 6 = 0 . 0 0 0 0 1 5
(b)
Est. (95% C.I.)SERBias%Est. (95% C.I.)SERBias%
𝑛 = 5 0
𝑔 3 : .8860 (.8808,.8917).0028−11.40 𝑡 3 : .1626 (.1618,.1634).0004−1.25
𝑔 4 : 1.021 (.9975,1.043).0116−31.93 𝑡 4 : .1314 (.1308,.1321).00030.12
𝑔 5 : 1.302 (1.218,1.389).0439−56.60 𝑡 5 : .0510 (.0505,.0516).0003−0.38
𝑔 6 : 1.623 (1.313,1.973).1673−78.36 𝑡 6 : .0484 (.0480,.0489).00020.14
𝑛 = 1 0 0
𝑔 3 : .9395 (.9350,.9436).0022−6.05 𝑡 3 : .1635 (.1629,.1640).0003−0.71
𝑔 4 : 1.225 (1.204,1.245).0104−18.33 𝑡 4 : .1314 (.1309,.1318).00020.09
𝑔 5 : 1.930 (1.847,2.032).0466−35.67 𝑡 5 : .0511 (.0507,.0514).0002−0.18
𝑔 6 : 3.385 (3.004,3.862).2160−54.87 𝑡 6 : .0483 (.0479,.0486).0002−0.07
𝑛 = 1 0 0 0
𝑔 3 : .9941 (.9926,.9957).0008−0.59 𝑡 3 : .1645 (.1643,.1647).0001−0.10
𝑔 4 : 1.471 (1.462,1.480).0046−1.93 𝑡 4 : .1313 (.1311,.1314).00010.05
𝑔 5 : 2.869 (2.822,2.921).0252−4.37 𝑡 5 : .0512 (.0510,.0512).00010.01
𝑔 6 : 6.889 (6.586,7.186).1544−8.15 𝑡 6 : .0483 (.0482,.0484).0001−0.01