lmomgld                package:lmomco                R Documentation

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_D_e_s_c_r_i_p_t_i_o_n:

     This function estimates the L-moments of the Generalized Lambda
     distribution given the parameters (xi, alpha, kappa, and h)  from
     'vec2par'. The L-moments in terms of the parameters are
     complicated; however, there are analytical solutions. There are no
     simple expressions of the parameters in terms of the L-moments.
     The first L-moment or the mean of the distribution is


  lambda_1 = xi + alpha (frac{1}{kappa+1} -  frac{1}{h+1} ) mbox{.}


     The second L-moment or L-scale in terms of the parameters and the
     mean is


 lambda_2 = xi + frac{2alpha}{(kappa+2)} -  2alpha ( frac{1}{h+1} -  frac{1}{h+2} ) - xi mbox{.}


     The third L-moment in terms of the parameters, the mean, and
     L-scale is 


 mbox{boldmath $Y$} = 2xi + frac{6alpha}{(kappa+3)} -  3(alpha+xi) + xi mbox{ and}


 lambda_3 = mbox{boldmath $Y$} + 6alpha (frac{2}{h+2} - frac{1}{h+3} -  frac{1}{h+1}) mbox{.}


     The fourth L-moment in termes of the parameters and the first
     three L-moments is


 mbox{boldmath $Y$} = frac{-3}{h+4}(frac{2}{h+2} -  frac{1}{h+3} -  frac{1}{h+1}) mbox{,}


 mbox{boldmath $Z$} = frac{20xi}{4} + frac{20alpha}{(kappa+4)} -  20 mbox{boldmath $Y$}alpha mbox{, and}


 lambda_4 = mbox{boldmath $Z$} -         5(kappa + 3(alpha+xi) - xi) + 6(alpha + xi) - xi mbox{.}


     It is conventional to express L-moments in terms of only the
     parameters and not the other L-moments. Lengthy algebra and
     further manipulation yields such a system of equations. The
     L-moments of the distribution are


  lambda_1 = xi + alpha (frac{1}{kappa+1} -  frac{1}{h+1} ) mbox{,}



 lambda_2 = alpha (frac{kappa}{(kappa+2)(kappa+1)} + frac{h}{(h+2)(h+1)}) mbox{,}



 lambda_3 =  alpha (frac{kappa (kappa - 1)} {(kappa+3)(kappa+2)(kappa+1)} - frac{h (h - 1)} {(h+3)(h+2)(h+1)} ) mbox{, and}



 lambda_4 = alpha (frac{kappa (kappa - 2)(kappa - 1)} {(kappa+4)(kappa+3)(kappa+2)(kappa+1)} + frac{h (h - 2)(h - 1)} {(h+4)(h+3)(h+2)(h+1)} ) mbox{.}


     The L-moment ratios are 


 tau_3 = frac{kappa(kappa-1)(h+3)(h+2)(h+1) -  h(h-1)(kappa+3)(kappa+2)(kappa+1)} {(kappa+3)(h+3) times [kappa(h+2)(h+1) +  h(kappa+2)(kappa+1)] } mbox{ and}



 tau_4 = frac{kappa(kappa-2)(kappa-1)(h+4)(h+3)(h+2)(h+1) +  h(h-2)(h-1)(kappa+4)(kappa+3)(kappa+2)(kappa+1)} {(kappa+4)(h+4)(kappa+3)(h+3) times [kappa(h+2)(h+1) +  h(kappa+2)(kappa+1)] } mbox{.}


     The pattern being established through symmetry, even higher
     L-moment ratios are readily obtained. Note the  alternating
     substraction and addition of the two terms in the numerator of the
     L-moment ratios (tau_r). For odd r >= 3 substraction is seen and
     for  even r >= 3 addition is seen. For example, the fifth L-moment
     ratio is


 N1 = kappa(kappa-3)(kappa-2)(kappa-1)(h+5)(h+4)(h+3)(h+2)(h+1) mbox{,}


 N2 = h(h-3)(h-2)(h-1)(kappa+5)(kappa+4)(kappa+3)(kappa+2)(kappa+1) mbox{,}


       D1 = (kappa+5)(h+5)(kappa+4)(h+4)(kappa+3)(h+3) mbox{,}


       D2 = [kappa(h+2)(h+1) + h(kappa+2)(kappa+1)] mbox{, and}


              tau_5 = frac{N1 - N2}{D1 times D2} mbox{.}


     By inspection the tau_r equations are not applicable for negative
     integer values k={-1, -2, -3, -4, ... } and h={-1, -2, -3, -4, ...
     } as division by zero will result. There are additional, but
     difficult to formulate, restrictions on the parameters both to
     define a valid Generalized Lambda distribution as well as valid
     L-moments. Verification of the parameters is conducted through
     'are.pargld.valid', and verification of the L-moment validity is
     conducted through 'are.lmom.valid'.

_U_s_a_g_e:

     lmomgld(gldpara)

_A_r_g_u_m_e_n_t_s:

 gldpara: The parameters of the distribution.

_V_a_l_u_e:

     An R 'list' is returned.

      L1: Arithmetic mean.

      L2: L-scale-analogous to standard deviation.

     LCV: coefficient of L-variation-analogous to coe. of variation.

    TAU3: The third L-moment ratio or L-skew-analogous to skew.

    TAU4: The fourth L-moment ratio or L-kurtosis-analogous to
          kurtosis.

    TAU5: The fifth L-moment ratio.

      L3: The third L-moment.

      L4: The fourth L-moment.

      L5: The fifth L-moment.

  source: An attribute identifying the computational  source of the
          L-moments: "lmomgld".

_A_u_t_h_o_r(_s):

     W.H. Asquith

_S_o_u_r_c_e:

     Derivations conducted by W.H. Asquith on February 11 and 12, 2006.

_R_e_f_e_r_e_n_c_e_s:

     Hosking, J.R.M., 1990, L-moments-Analysis and estimation of
     distributions using linear combinations of order statistics:
     Journal of the Royal Statistical Society, Series B, vol. 52, p.
     105-124.

     Karvanen, J., Eriksson, J., and Koivunen, V., 2002, Adaptive score
     functions for maximum likelihood ICA: Journal of VLSI Signal
     Processing, vol. 32, p. 82-92.

     Karian, Z.A., and Dudewicz, E.J., 2000, Fitting statistical
     distibutions-The generalized lambda distribution and generalized
     bootstrap methods:  CRC Press, Boca Raton, FL, 438 p.

_S_e_e _A_l_s_o:

     'pargld, \code{cdfgld}, \code{quagld}'

_E_x_a_m_p_l_e_s:

     lmomgld(vec2par(c(10,10,0.4,1.3),type='gld'))

