Approximations for use in constructing L-moment ratio diagrams

To construct an L-moment ratio diagram it is convenient to have simple explicit expressions for tau_4 (L-kurtosis) in terms of tau_3 (L-skewness) for some commonly used probability distribtions. Polynomial approximations of the form

                                           2                     8
 tau_4  = A0  + A1 * (tau_3) + A2 * (tau_3)  + ... + A8 * (tau_3)

have been obtained, and the coefficients are given in the table below. "Overall lower bound" is the lower bound on tau_4 for all distributions [Hosking, J. R. Statist. Soc. B, 1990, eq.(2.7)]. For given tau_3, the approximations yield values of tau_4 that are accurate to within 0.0005 provided that tau_3 is in the range -0.9 to +0.9, except that for the generalized extreme-value distribution 0.0005 accuracy is attained only when tau_3 is between -0.6 and +0.9,

The approximations are not intended for detailed analytical calculations -- for that purpose, use the routines in the LMOMENTS software package -- but they are sufficiently accurate for use in plotting theoretical L-moment relationships on an L-moment ratio diagram.

-------------------------------------------------------------------------------
       Generalized  Generalized   Generalized             Pearson    Overall
        logistic   extreme-value    Pareto     Lognormal type III  lower bound
-------------------------------------------------------------------------------
 A0      0.16667      0.10701       0.          0.12282   0.12240    -0.25
 A1       .           0.11090       0.20196      .         .           .
 A2      0.83333      0.84838       0.95924     0.77518   0.30115     1.25
 A3       .          -0.06669      -0.20096      .         .           .
 A4       .           0.00567       0.04061     0.12279   0.95812      .
 A5       .          -0.04208        .           .         .           .
 A6       .           0.03673        .         -0.13638  -0.57488      .
 A7       .            .             .           .         .           .
 A8       .            .             .          0.11368   0.19383      .
-------------------------------------------------------------------------------

This material is based on an IBM Research Report by J. R. M. Hosking ["Approximations for use in constructing L-moment ratio diagrams", Research Report RC 16635, IBM Research Division, Yorktown Heights, N.Y., 1991]. The approximations are also given in Appendix A.12 of the book Regional frequency analysis: an approach based on L-moments, by J. R. M. Hosking and J. R. Wallis.

Last modified: 1 March 1999.

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