ldTweedie                package:mgcv                R Documentation

_L_o_g _T_w_e_e_d_i_e _d_e_n_s_i_t_y _e_v_a_l_u_a_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

     A function to evaluate the log of the Tweedie density for variance
     powers between 1 and 2, inclusive. Also evaluates first and second
     derivatives of log density w.r.t. its scale parameter.

_U_s_a_g_e:

     ldTweedie(y,mu=y,p=1.5,phi=1)

_A_r_g_u_m_e_n_t_s:

       y: values at which to evaluate density.

      mu: corresponding means (either of same length as 'y' or a single
          value).

       p: the variance of 'y' is proportional to its mean to the power
          'p'. 'p' must be between 1 and 2. 1 is Poisson like (exactly
          Poisson if 'phi=1'), 2 is gamma. 

     phi: The scale parameter. Variance of 'y' is 'phi*mu^p'.

_D_e_t_a_i_l_s:

     A Tweedie random variable with 1<p<2 is a sum of 'N' gamma random
     variables  where 'N' has a Poisson distribution. The p=1 case is a
     generalization of a Poisson distribution and is a discrete 
     distribution supported on integer multiples of the scale
     parameter. For 1<p<2 the distribution is supported on the 
     positive reals with a point mass at zero. p=2 is a gamma
     distribution. As p gets very close to 1 the continuous 
     distribution begins to converge on the discretely supported limit
     at p=1. 

     'ldTweedie' is based on the series evaluation method of Dunn and
     Smyth (2005). Without the restriction on 'p' the calculation of
     Tweedie densities is less straightforward. If you really need this
      case then the 'tweedie' package is the place to start.

_V_a_l_u_e:

     A matrix with 3 columns. The first is the log density of 'y' (log
     probability if 'p=1').  The second and third are the first and
     second derivatives of the log density w.r.t. 'phi'.

_A_u_t_h_o_r(_s):

     Simon N. Wood simon.wood@r-project.org modified from Venables and
     Ripley's 'negative.binomial' family.

_R_e_f_e_r_e_n_c_e_s:

     Dunn, P.K. and G.K. Smith (2005) Series evaluation of Tweedie
     exponential dispersion model densities.  Statistics and Computing
     15:267-280

     Tweedie, M. C. K. (1984). An index which distinguishes between
     some important exponential families. Statistics: Applications and
     New Directions. Proceedings of the Indian Statistical Institute
     Golden Jubilee International Conference (Eds. J. K. Ghosh and J.
     Roy), pp. 579-604. Calcutta: Indian Statistical Institute.

_E_x_a_m_p_l_e_s:

       library(mgcv)
       ## convergence to Poisson illustrated
       ## notice how p>1.1 is OK
       y <- seq(1e-10,10,length=1000)
       p <- c(1.0001,1.001,1.01,1.1,1.2,1.5,1.8,2)
       phi <- .5
       fy <- exp(ldTweedie(y,mu=2,p=p[1],phi=phi)[,1])
       plot(y,fy,type="l",ylim=c(0,3),main="Tweedie density as p changes")
       for (i in 2:length(p)) {
         fy <- exp(ldTweedie(y,mu=2,p=p[i],phi=phi)[,1])
         lines(y,fy,col=i)
       }

