laplaceUC                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 Laplace distribution with location parameter
     'location' and scale parameter 'scale'.

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

     dlaplace(x, location=0, scale=1)
     plaplace(q, location=0, scale=1)
     qlaplace(p, location=0, scale=1)
     rlaplace(n, location=0, scale=1)

_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 a, which is the mean. 

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

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

     The Laplace distribution is often known as the double-exponential
     distribution and, for modelling, has heavier tail than the normal
     distribution. The Laplace density function is 

                   f(y) =  (1/(2b)) exp( -|y-a|/b )

     where -Inf<y<Inf, -Inf<a<Inf and b>0. The mean is a and the
     variance is 2b^2. 

     See 'laplace', the 'VGAM' family function for estimating the two
     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:

     'dlaplace' gives the density, 'plaplace' gives the distribution
     function, 'qlaplace' gives the quantile function, and 'rlaplace'
     generates random deviates.

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

     T. W. Yee

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

     Evans, M., Hastings, N. and Peacock, B. (2000) _Statistical
     Distributions_, New York: Wiley-Interscience, Third edition.

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

     'laplace'.

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

     loc = 1; b = 2
     y = rlaplace(n=100, loc=loc, scale=b)
     mean(y)
     loc      # population mean
     var(y)
     2 * b^2  # population variance

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

     plaplace(qlaplace(seq(0.05,0.95,by=0.05), loc, b), loc, b)

