Weibull                 package:Rlab                 R Documentation

_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:

     Density, distribution function, quantile function and random
     generation for the Weibull distribution with parameters 'alpha'
     (or 'shape') and 'beta' (or 'scale').

     This special Rlab implementation allows the parameters 'alpha' and
     'beta' to be used, to match the function description often found
     in textbooks.

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

     dweibull(x, shape, scale = 1, alpha = shape, beta = scale, log = FALSE)
     pweibull(q, shape, scale = 1, alpha = shape, beta = scale,
              lower.tail = TRUE, log.p = FALSE)
     qweibull(p, shape, scale = 1, alpha = shape, beta = scale,
              lower.tail = TRUE, log.p = FALSE)
     rweibull(n, shape, scale = 1, alpha = shape, beta = scale)

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

    x, q: vector of quantiles.

       p: vector of probabilities.

       n: number of observations. If 'length(n) > 1', the length is
          taken to be the number required.

shape, scale: shape and scale parameters, the latter defaulting to 1.

alpha, beta: alpha and beta parameters, the latter defaulting to 1.

log, log.p: logical; if TRUE, probabilities p are given as log(p).

lower.tail: logical; if TRUE (default), probabilities are P[X <= x],
          otherwise, P[X > x].

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

     The Weibull distribution with 'alpha' (or 'shape') parameter a and
     'beta' (or 'scale') parameter b has density given by

               f(x) = (a/b) (x/b)^(a-1) exp(- (x/b)^a)

     for x > 0. The cumulative is F(x) = 1 - exp(- (x/b)^a), the mean
     is E(X) = b Gamma(1 + 1/a), and the Var(X) = b^2 * (gamma(1 + 2/a)
     - (gamma(1 + 1/a))^2).

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

     'dweibull' gives the density, 'pweibull' gives the distribution
     function, 'qweibull' gives the quantile function, and 'rweibull'
     generates random deviates.

_N_o_t_e:

     The cumulative hazard H(t) = - log(1 - F(t)) is '-pweibull(t, a,
     b, lower = FALSE, log = TRUE)' which is just H(t) = {(t/b)}^a.

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

     'dexp' for the Exponential which is a special case of a Weibull
     distribution.

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

     x <- c(0,rlnorm(50))
     all.equal(dweibull(x, alpha = 1), dexp(x))
     all.equal(pweibull(x, alpha = 1, beta = pi), pexp(x, rate = 1/pi))
     ## Cumulative hazard H():
     all.equal(pweibull(x, 2.5, pi, lower=FALSE, log=TRUE), -(x/pi)^2.5, tol=1e-15)
     all.equal(qweibull(x/11, alpha = 1, beta = pi), qexp(x/11, rate = 1/pi))

