bisa                  package:VGAM                  R Documentation

_B_i_r_n_b_a_u_m-_S_a_u_n_d_e_r_s _D_i_s_t_r_i_b_u_t_i_o_n _F_a_m_i_l_y _F_u_n_c_t_i_o_n

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

     Estimates the shape and scale parameters of the Birnbaum-Saunders
     distribution by maximum likelihood estimation.

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

     bisa(lshape = "loge", lscale = "loge",
          eshape = list(), escale = list(),
          ishape = NULL, iscale = 1, method.init = 1, zero = NULL)

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

lscale, lshape: Parameter link functions applied to the shape and scale
          parameters (a and b below). See 'Links' for more choices. A
          log link is the default for both because they are positive.

escale, eshape: List. Extra argument for each of the links. See 'earg'
          in 'Links' for general information.

iscale, ishape: Initial values for a and b. A 'NULL' means an initial
          value is chosen internally using 'method.init'.

method.init: An integer with value '1' or '2' or '3' which specifies
          the initialization method. If failure to converge occurs try
          the other value, or else specify a value for  'ishape' and/or
          'iscale'. 

    zero: An integer-valued vector specifying which linear/additive
          predictors are modelled as intercepts only. The default is
          none of them. If used, choose one value from the set {1,2}.

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

     The (two-parameter) Birnbaum-Saunders distribution  has a
     cumulative distribution function that can be written as

                     F(y;a,k) = pnorm[xi(y/b)/a]

     where pnorm() is the  cumulative distribution function of a
     standard normal (see 'pnorm'), xi(t) = t^(0.5) - t^(-0.5), y > 0,
     a>0 is the shape parameter, b>0 is the scale parameter. The mean
     of Y (which is the fitted value) is b*(1 + a*a/2). and the
     variance is a^2 b^2 (1 + (5/4)*a^2). By default, eta1=log(a) and
     eta2=log(b) for this family function.

     Note that a and b are orthogonal, i.e., the Fisher information
     matrix is diagonal. This family function implements Fisher
     scoring, and it is unnecessary to compute any integrals
     numerically.

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

     An object of class '"vglmff"' (see 'vglmff-class'). The object is
     used by modelling functions such as 'vglm', and 'vgam'.

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

     T. W. Yee

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

     Lemonte, A. J. and Cribari-Neto, F. and Vasconcellos, K. L. P.
     (2007) Improved statistical inference for the two-parameter
     Birnbaum-Saunders distribution. _Computational Statistics & Data
     Analysis_, *51*, 4656-4681.

     Birnbaum, Z. W. and Saunders, S. C. (1969) A new family of life
     distributions. _Journal of Applied Probability_, *6*, 319-327.

     Birnbaum, Z. W. and Saunders, S. C. (1969) Estimation for a family
     of life distributions with applications to fatigue. _Journal of
     Applied Probability_, *6*, 328-347.

     Engelhardt, M. and Bain, L. J. and Wright, F. T. (1981) Inferences
     on the parameters of the Birnbaum-Saunders fatigue life
     distribution based on maximum likelihood estimation.
     _Technometrics_, *23*, 251-256.

     Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1995)
     _Continuous Univariate Distributions_, 2nd edition, Volume 2, New
     York: Wiley.

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

     'pbisa', 'inv.gaussianff'.

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

     x = runif(n <- 1000)
     y = rbisa(n, shape=exp(-0.5+x), scale=exp(1.5))
     fit = vglm(y ~ x, bisa(zero=2), trace=TRUE)
     coef(fit, matrix=TRUE)

     ## Not run: 
     y = rbisa(n=1000, shape=exp(-0.5), scale=exp(0.5))
     fit = vglm(y ~ 1, bisa, trace=TRUE)
     hist(y, prob=TRUE, ylim=c(0,0.5), col="lightblue")
     coef(fit, matrix=TRUE)
     mean(y)
     fitted(fit)[1:4]
     x = seq(0, max(y), len=200)
     lines(x, dbisa(x, Coef(fit)[1], Coef(fit)[2]), col="red", lwd=2)
     ## End(Not run)

