lmomTLgld               package:lmomco               R Documentation

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

     This function estimates the symmetrical trimmed L-moments
     (TL-moments) for t=1 of the Generalized Lambda distribution given
     the parameters (xi, alpha, kappa, and h) from 'vec2par'. The
     TL-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 four
     TL-moments of the distribution are


 lambda^{(1)}_1 = xi + 6alpha (frac{1}{(kappa+3)(kappa+2)} -  frac{1}{(h+3)(h+2)} ) mbox{,}



 lambda^{(1)}_2 = 6alpha (frac{kappa}{(kappa+4)(kappa+3)(kappa+2)} + frac{h}{(h+4)(h+3)(h+2)}) mbox{,}



 lambda^{(1)}_3 =  frac{20alpha}{3} (frac{kappa (kappa - 1)} {(kappa+5)(kappa+4)(kappa+3)(kappa+2)} - frac{h (h - 1)} {(h+5)(h+4)(h+3)(h+2)} ) mbox{,}



 lambda^{(1)}_4 = frac{15alpha}{2} (frac{kappa (kappa - 2)(kappa - 1)} {(kappa+6)(kappa+5)(kappa+4)(kappa+3)(kappa+2)} + frac{h (h - 2)(h - 1)} {(h+6)(h+5)(h+4)(h+3)(h+2)} ) mbox{, and}


         lambda^{(1)}_5 = frac{42alpha}{5} (N1 - N2 ) mbox{,}


     where


 N1 = frac{kappa (kappa - 3)(kappa - 2)(kappa - 1) } {(kappa+7)(kappa+6)(kappa+5)(kappa+4)(kappa+3)(kappa+2)} mbox{ and}


 N2 = frac{h (h - 3)(h - 2)(h - 1)}{(h+7)(h+6)(h+5)(h+4)(h+3)(h+2)} mbox{.}


     The TL-moment (t=1) for tau^{(1)}_3 is 


 tau^{(1)}_3 = frac{10}{9} ( frac{kappa(kappa-1)(h+5)(h+4)(h+3)(h+2) -  h(h-1)(kappa+5)(kappa+4)(kappa+3)(kappa+2)} {(kappa+5)(h+5) times [kappa(h+4)(h+3)(h+2) +  h(kappa+4)(kappa+3)(kappa+2)] } ) mbox{.}


     The TL-moment (t=1) for tau^{(1)}_4 is 


    N1 = kappa(kappa-2)(kappa-1)(h+6)(h+5)(h+4)(h+3)(h+2) mbox{,}


 N2 = h(h-2)(h-1)(kappa+6)(kappa+5)(kappa+4)(kappa+3)(kappa+2) mbox{,}


              D1 = (kappa+6)(h+6)(kappa+5)(h+5) mbox{,}


 D2 = [kappa(h+4)(h+3)(h+2) + h(kappa+4)(kappa+3)(kappa+2)] mbox{, and}


   tau^{(1)}_4 = frac{5}{4} ( frac{N1 +  N2}{D1 times D2} ) mbox{.}


     Finally the TL-moment (t=1) for tau^{(1)}_5 is


 N1 = kappa(kappa-3)(kappa-2)(kappa-1)(h+7)(h+6)(h+5)(h+4)(h+3)(h+2) mbox{,}


 N2 = h(h-3)(h-2)(h-1)(kappa+7)(kappa+6)(kappa+5)(kappa+4)(kappa+3)(kappa+2) mbox{,}


       D1 = (kappa+7)(h+7)(kappa+6)(h+6)(kappa+5)(h+5) mbox{,}


 D2 = [kappa(h+4)(h+3)(h+2) + h(kappa+4)(kappa+3)(kappa+2)] mbox{, and}


   tau^{(1)}_5 = frac{7}{5} ( frac{N1 -  N2}{D1 times D2} )mbox{.}


     By inspection the tau_r equations are not applicable for negative
     integer values k={-2, -3, -4, ... } and h={-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:

     lmomTLgld(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.

 lambdas: Vector of the TL-moments. First element is lambda^{(1)}_1,
          second element is lambda^{(1)}_2, and so on.

  ratios: Vector of the TL-moment ratios. Second element is  tau^{(1)},
          third element is tau^{(1)}_3 and so on. 

    trim: Trim level = 1

  source: An attribute identifying the computational source  of the
          TL-moments: "lmomTLgld".

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

     W.H. Asquith

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

     Derivations conducted by W.H. Asquith on February 18 and 19, 2006.

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

     Elamir, E.A.H., and Seheult, A.H., 2003, Trimmed L-moments:
     Computational statistics and data analysis, vol. 43, pp. 299-314.

     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.

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

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

     'lmomgld', 'pargld', 'cdfgld', 'quagld'

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

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

