cdfwei                package:lmomco                R Documentation

_C_u_m_u_l_a_t_i_v_e _D_i_s_t_r_i_b_u_t_i_o_n _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 cumulative probability or nonexceedance
     probability of the Weibull distribution given parameters (zeta,
     beta, and delta) of the distribution computed by 'parwei'. The
     cumulative distribution function of the distribution is


          F(x) = 1 - e^{-{frac{x+zeta}{beta}}^delta} mbox{,}


     where F(x) is the nonexceedance probability for quantile x, 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 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 cumulative distribution function of the Weibull
     distribution is 'pweibull'. Given a Weibull parameter object
     'para', the R syntax is
     'pweibull(x+para$para[1],para$para[3],scale=para$para[2])'. For
     the current implementation for this package, the reversed
     Generalized Extreme Value distribution is used
     '1-cdfgev(-x,para)'.

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

     cdfwei(x, para)

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

       x: A real value.

    para: The parameters from 'parwei' or similar.

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

     Nonexceedance probability (F) for x.

_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:

     'quawei', 'parwei'

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

       # Evaluate Weibull deployed here and within R (pweibull)
       lmr <- lmom.ub(c(123,34,4,654,37,78))
       WEI <- parwei(lmr)
       F1  <- cdfwei(50,WEI)
       F2  <- pweibull(50+WEI$para[1],shape=WEI$para[3],scale=WEI$para[2])
       if(F1 == F2) 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(cdfwei(Q,WEI),Q,lwd=5,col=8)
       lines(1-cdfgev(-Q,GEV),Q,col=2) # line over laps previous

