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",
          ishape = NULL, iscale = 1, method.init = 1,
          fsmax=9001, 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.

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' which specifies the
          initialization method. If failure to converge occurs try the
          other value, or else specify a value for  'ishape' and/or
          'iscale'. 

   fsmax: Integer. If the formula is an intercept-only or if the number
          of observations n is less than 'fsmax' then Fisher scoring is
          used (recommended), else a BFGS quasi-Newton update formula
          for the working weight matrices is used.

    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.

_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'.

_N_o_t_e:

     If the formula is an intercept-only or n is sufficiently small,
     then this family function implements Fisher scoring.  This
     involves computing an integral numerically. Fisher scoring is
     generally recommended here provided the integrals can be computed
     successfully and it does not take too long.

     For n large and non-intercept-only formulas the BFGS quasi-Newton
     update formula for the working weight matrices is used by default.
     This is more numerically fraught. Additionally, the estimated
     variance-covariance matrix may be inaccurate or simply wrong! The
     standard errors must be therefore treated with caution; these are
     computed in functions such as 'vcov()' and 'summary()'.

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

     T. W. Yee

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

     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:

     y = rbisa(n=1000, shape=exp(-0.5), scale=exp(0.5))
     fit1 = vglm(y ~ 1, bisa, trace=TRUE)
     coef(fit1, matrix=TRUE)
     mean(y)
     fitted(fit1)[1:4]

     ## Not run: 
     hist(y, prob=TRUE)
     x = seq(0, max(y), len=200)
     lines(x, dbisa(x, Coef(fit1)[1], Coef(fit1)[2]), col="red")
     ## End(Not run)

