Tweedie                 package:mgcv                 R Documentation

_G_A_M _T_w_e_e_d_i_e _f_a_m_i_l_y

_D_e_s_c_r_i_p_t_i_o_n:

     A Tweedie family, designed for use with 'gam' from the 'mgcv'
     library. Restricted to variance function powers between 1 and 2. A
     useful alternative to 'quasi' when a full likelihood is desirable.

_U_s_a_g_e:

     Tweedie(p=1, link = power(0))

_A_r_g_u_m_e_n_t_s:

       p: the variance of an observation is proportional to its mean to
          the power 'p'. 'p' must be greater than 1 and less than or
          equal to 2. 1 would be Poisson, 2 is gamma. 

    link: The link function: one of '"log"', '"identity"', '"inverse"',
          '"sqrt"', or a  'power' link.

_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, and is therefore highly multimodal.  See 'ldTweedie' for
     more on this behaviour.

     'Tweedie' is based partly on the 'poisson' family, and partly on
     'tweedie' from the  'statmod' package. It includes extra
     components to work with all 'mgcv' GAM fitting methods as well as 
     an 'aic' function. The required log density evaluation (+
     derivatives w.r.t. scale) 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, and there does not  currently
     seem to be an implementation which offers any benefit over
     'quasi'. If you really need to this  case then the 'tweedie'
     package is the place to start.

_V_a_l_u_e:

     An object inheriting from class 'family', with additional elements 

    dvar: the function giving the first derivative of the variance
          function w.r.t. 'mu'.

   d2var: the function giving the second derivative of the variance
          function w.r.t. 'mu'.

      ls: A function returning a 3 element array: the saturated log
          likelihood followed by its first 2 derivatives w.r.t. the
          scale parameter.

_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.

_S_e_e _A_l_s_o:

     'ldTweedie'

_E_x_a_m_p_l_e_s:

     library(mgcv)
     set.seed(3)
     n<-400
     ## Simulate data (really Poisson with log-link)
     dat <- gamSim(1,n=n,dist="poisson",scale=.2)

     ## Fit a `nearby' Tweedie...
     b <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=Tweedie(1.1,power(.1)),
              data=dat)
     plot(b,pages=1)
     print(b)

     ## Same by approximate REML...
     b1 <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=Tweedie(1.1,power(.1)),
               data=dat,method="REML")
     plot(b1,pages=1)
     print(b1)

     rm(dat)

