alaplaceUC               package:VGAM               R Documentation

_T_h_e _L_a_p_l_a_c_e _D_i_s_t_r_i_b_u_t_i_o_n

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

     Density, distribution function, quantile function and random
     generation for the 3-parameter asymmetric Laplace distribution
     with location parameter 'location', scale parameter 'scale', and
     asymmetry parameter 'kappa'.

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

     dalaplace(x, location=0, scale=1, tau=0.5, kappa=sqrt(tau/(1-tau)))
     palaplace(q, location=0, scale=1, tau=0.5, kappa=sqrt(tau/(1-tau)))
     qalaplace(p, location=0, scale=1, tau=0.5, kappa=sqrt(tau/(1-tau)))
     ralaplace(n, location=0, scale=1, tau=0.5, kappa=sqrt(tau/(1-tau)))

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

    x, q: vector of quantiles.

       p: vector of probabilities.

       n: number of observations. Positive integer of length 1.

location: the location parameter xi. 

   scale: the scale parameter b. Must consist of positive values. 

     tau: the quantile parameter tau. Must consist of values in (0,1).
          This argument is used to specify 'kappa' and is ignored if
          'kappa' is assigned.

   kappa: the asymmetry parameter kappa. Must consist of positive
          values. 

_D_e_t_a_i_l_s:

     There are several variants of asymmetric Laplace distributions
     (ALDs) and this one is known as _the_ ALD by Kotz et al. (2001).
     See 'alaplace3', the 'VGAM' family function for estimating the
     three parameters by maximum likelihood estimation, for formulae
     and details. Apart from 'n', all the above arguments may be
     vectors and are recyled to the appropriate length if necessary.

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

     'dalaplace' gives the density, 'palaplace' gives the distribution
     function, 'qalaplace' gives the quantile function, and 'ralaplace'
     generates random deviates.

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

     T. W. Yee

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

     Kotz, S., Kozubowski, T. J. and Podgorski, K. (2001) _The Laplace
     distribution and generalizations: a revisit with applications to
     communications, economics, engineering, and finance_, Boston:
     Birkhauser.

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

     'alaplace3'.

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

     x = seq(-5, 5, by=0.01)
     loc = 0; b = 1.5; kappa = 2
     ## Not run: 
     plot(x, dalaplace(x, loc, b, kappa=kappa), type="l", col="blue",
          main="Blue is density, red is cumulative distribution function",
          ylim=c(0,1), sub="Purple are 5,10,...,95 percentiles", las=1, ylab="")
     abline(h=0, col="blue", lty=2)
     lines(qalaplace(seq(0.05,0.95,by=0.05), loc, b, kappa=kappa),
           dalaplace(qalaplace(seq(0.05,0.95,by=0.05), loc, b, kappa=kappa),
                     loc, b, kappa=kappa), col="purple", lty=3, type="h")
     lines(x, palaplace(x, loc, b, kappa=kappa), type="l", col="red")
     abline(h=0, lty=2)
     ## End(Not run)
     palaplace(qalaplace(seq(0.05,0.95,by=0.05), loc, b, kappa=kappa),
               loc, b, kappa=kappa)

