zeta                   package:sn                   R Documentation

_F_u_n_c_t_i_o_n _l_o_g(_2*_p_n_o_r_m(_x)) _a_n_d _i_t_s _d_e_r_i_v_a_t_i_v_e_s

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

     The function 'log(2*(pnorm(x))' and its derivatives,  including
     inverse Mills ratio.

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

     zeta(k, x)

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

       k: an integer scalar between 0 and 5. 

       x: a vector. Missing values ('NA's)  and 'Inf's are allowed 

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

     For 'k' between 0 and 5, the derivative of  order 'k' of
     'log(2*pnorm(x))' is  evaluated;   the derivative of order 'k=0'
     refers to  the function itself. If 'k' is not integer, it is
     converted to integer and a warning message is generated. If 'k<0'
     or 'k>5',  'NULL' is returned.

     The computation for 'k>1' is reduced to the case 'k=1', making use
     of expressions given by Azzalini and Capitanio (1999).  For
     numerical stability, the evaluation of 'zeta(1,x)' when 'x < -50'
     makes use of the asymptotic expansion (26.2.13) in Abramowitz and
     Stegun (1964).

     'zeta(1,-x)' equals 'dnorm(x)/pnorm(-x)' (in principle, apart from
     the asymptotic expansion mentioned above), called the   _inverse
     Mills ratio_.

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

     a vector giving the 'k'-th order derivative evaluated at 'x'

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

     Abramowitz, M. and Stegun, I. A., editors (1964). _Handbook of
     Mathematical Functions_.  Dover Publications.

     Azzalini, A. and Capitanio, A. (1999). Statistical applications of
     the multivariate skew-normal distribution. Technical report
     available at <URL: http://azzalini.stat.unipd.it/SN>. An abriged
     version is published in   _J.Roy.Statist.Soc. B_ *61*, 579-602.

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

     y <- zeta(2,seq(-20,20,by=0.5))
     #
     for(k in 0:5) curve(zeta(k,x), from=-1.5, to=5, col = k+2, add = k > 0)
     legend(3.5, -0.5, legend=as.character(0:5), col=2:7, lty=1)

