quawei                package:lmomco                R Documentation

_Q_u_a_n_t_i_l_e _F_u_n_c_t_i_o_n _o_f _t_h_e _W_e_i_b_u_l_l _D_i_s_t_r_i_b_u_t_i_o_n

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

     This function computes the quantiles of the Weibull distribution
     given parameters (zeta, beta, and delta) of the distribution
     computed by 'parwei'. The quantile function of the distribution is


           x(F) = beta[- log(1-F)]^{1/delta} - zeta mbox{,}


     where x(F) is the quantile for nonexceedance probability F, zeta
     is a location parameter, beta is a scale parameter, and delta is a
     shape parameter.

     The Weibull distribution is a reverse Generalized Extreme Value
     distribution.  As result, the Generalized Extreme Value algorithms
     are used for implementation of the Weibull in this package. The
     relation between the Generalized Extreme Value distribution
     parameters (xi, alpha, and kappa) is


                       kappa = 1/delta mbox{,}



                    alpha = beta/delta mbox{, and}



                       xi = zeta - beta mbox{.}


     These relations are taken from Hosking and Wallis (1997).

     In R the quantile function of the Weibull distribution is
     'qweibull'. Given a Weibull parameter object 'p', the R syntax is
     'qweibull(f, p$para[3], scale=p$para[2]) - p$para[1]'. For the
     current implementation for this package, the reversed Generalized
     Extreme Value distribution is used '-quagev((1-f),para)'.

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

     quawei(f, para)

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

       f: Nonexceedance probability (0 <= F <= 1).

    para: The parameters from 'parwei' or similar.

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

     Quantile value for nonexceedance probability F.

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

     W.H. Asquith

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

     Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency
     analysis-An approach based on L-moments: Cambridge University
     Press.

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

     'cdfwei', 'parwei'

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

       # Evaluate Weibull deployed here and within R (qweibull)
       lmr <- lmom.ub(c(123,34,4,654,37,78))
       WEI <- parwei(lmr)
       Q1  <- quawei(0.5,WEI)
       Q2  <- qweibull(0.5,shape=WEI$para[3],scale=WEI$para[2])-WEI$para[1]
       if(Q1 == Q2) EQUAL <- TRUE

       # The Weibull is a reversed generalized extreme value
       Q <- sort(rlmomco(34,WEI)) # generate Weibull sample
       lm1 <- lmoms(Q)    # regular L-moments
       lm2 <- lmoms(-Q)   # L-moment of negated (reversed) data
       WEI <- parwei(lm1) # parameters of Weibull
       GEV <- pargev(lm2) # parameters of GEV
       F <- nonexceeds()  # Get a vector of nonexceedance probs
       plot(pp(Q),Q) 
       lines(F,quawei(F,WEI))
       lines(F,-quagev(1-F,GEV),col=2) # line over laps previous

