skewnormal1               package:VGAM               R Documentation

_U_n_i_v_a_r_i_a_t_e _S_k_e_w-_N_o_r_m_a_l _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:

     Maximum likelihood estimation of the shape parameter of a
     univariate skew-normal distribution.

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

     skewnormal1(lshape = "identity", ishape = NULL)

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

  lshape: Link function applied to the shape parameter. See 'Links' for
          more choices.

  ishape: Optional inital value for the shape parameter. The default is
          to choose one internally. See the note below.

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

     The univariate skew-normal distribution has a density function
     that can be written

                  f(y) = 2 * phi(y) * Phi(alpha * y)

     where alpha is the shape parameter. Here, phi is the standard
     normal density and Phi its cumulative distribution function. When
     alpha=0 the result is a standard normal distribution. When alpha=1
     it models the distribution of the maximum of two independent
     standard normal variates. When the absolute value of the shape
     parameter increases the skewness of the distribution increases.
     The limit as the shape parameter tends to positive infinity
     results in the folded normal distribution or half-normal
     distribution. When the shape parameter changes its sign, the
     density is reflected about y=0.

     The mean of the distribution is mu=alpha*sqrt(2/(pi*(1+alpha^2)))
     and these are returned as the fitted values. The variance of the
     distribution is 1-mu^2. The Newton-Raphson algorithm is used.

_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:

     It is a good idea to use several different initial values to
     ensure that the global solution is obtained.

     This family function will be modified (hopefully soon) to handle a
     location and scale parameter too.

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

     Thomas W. Yee

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

     Azzalini, A. A. (1985). A class of distributions which include the
     normal. _Scandinavian Journal of Statistics_, *12*, 171-178.

     Azzalini, A. and Capitanio, A. (1999). Statistical applications of
     the multivariate skew-normal distribution. _Journal of the Royal
     Statistical Society, Series B, Methodological_, *61*, 579-602.

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

     'snorm', 'normal1'.

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

     y = rsnorm(n <- 1000, shape=5)
     fit = vglm(y ~ 1, skewnormal1, trace=TRUE)
     coef(fit, matrix=TRUE)
     fitted(fit)[1:4,]
     mean(y)
     ## Not run: 
     hist(y, prob=TRUE)
     x = seq(min(y), max(y), len=200)
     lines(x, dsnorm(x, shape=Coef(fit)), col="blue")
     ## End(Not run)

     x = runif(n)
     y = rsnorm(n, shape=1 + 2*x)
     fit = vglm(y ~ x, skewnormal1, trace=TRUE, crit="coef")
     summary(fit)

