\HeaderA{ldTweedie}{Log Tweedie density evaluation}{ldTweedie}
\keyword{models}{ldTweedie}
\keyword{regression}{ldTweedie}
\begin{Description}\relax
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.
\end{Description}
\begin{Usage}
\begin{verbatim}
ldTweedie(y,mu=y,p=1.5,phi=1)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{y}] values at which to evaluate density.
\item[\code{mu}] corresponding means (either of same length as \code{y} or a single value).
\item[\code{p}] the variance of \code{y} is proportional to its mean to the power \code{p}. \code{p} must
be between 1 and 2. 1 is Poisson like (exactly Poisson if \code{phi=1}), 2 is gamma. 
\item[\code{phi}] The scale parameter. Variance of \code{y} is \code{phi*mu\textasciicircum{}p}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
A Tweedie random variable with 1\textless{}p\textless{}2 is a sum of \code{N} gamma random variables 
where \code{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\textless{}p\textless{}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. 

\code{ldTweedie} is based on the series evaluation method of Dunn and Smyth (2005). Without the restriction on \code{p} the calculation of Tweedie densities is less straightforward. If you really need this 
case then the \code{tweedie} package is the place to start.
\end{Details}
\begin{Value}
A matrix with 3 columns. The first is the log density of \code{y} (log probability if \code{p=1}). 
The second and third are the first and second derivatives of the log density w.r.t. \code{phi}.
\end{Value}
\begin{Author}\relax
Simon N. Wood \email{simon.wood@r-project.org}
modified from Venables and Ripley's \code{negative.binomial} family.
\end{Author}
\begin{References}\relax
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.
\end{References}
\begin{Examples}
\begin{ExampleCode}
  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)
  }

\end{ExampleCode}
\end{Examples}

