# dtweedie.dldphi

0th

Percentile

##### Tweedie Distributions

Derivatives of the log-likelihood with respect to $\phi$

Keywords
models
##### Usage
dtweedie.dldphi(phi, mu, power, y )
dtweedie.dldphi.saddle(phi, mu, power, y )
##### Arguments
y

vector of quantiles

mu

the mean

phi

the dispersion

power

the value of $p$ such that the variance is $\mbox{var}[Y]=\phi\mu^p$

##### Details

The Tweedie family of distributions belong to the class of exponential dispersion models (EDMs), famous for their role in generalized linear models. The Tweedie distributions are the EDMs with a variance of the form $\mbox{var}[Y]=\phi\mu^p$ where $p$ is greater than or equal to one, or less than or equal to zero. This function only evaluates for $p$ greater than or equal to one. Special cases include the normal ($p=0$), Poisson ($p=1$ with $\phi=1$), gamma ($p=2$) and inverse Gaussian ($p=3$) distributions. For other values of power, the distributions are still defined but cannot be written in closed form, and hence evaluation is very difficult.

##### Value

the value of the derivative $\partial\ell/\partial\phi$ where $\ell$ is the log-likelihood for the specified Tweedie distribution. dtweedie.dldphi.saddle uses the saddlepoint approximation to determine the derivative; dtweedie.dldphi uses an infinite series expansion.

##### References

Dunn, P. K. and Smyth, G. K. (2008). Evaluation of Tweedie exponential dispersion model densities by Fourier inversion. Statistics and Computing, 18, 73--86. 10.1007/s11222-007-9039-6

Dunn, Peter K and Smyth, Gordon K (2005). Series evaluation of Tweedie exponential dispersion model densities Statistics and Computing, 15(4). 267--280. 10.1007/s11222-005-4070-y

Dunn, Peter K and Smyth, Gordon K (2001). Tweedie family densities: methods of evaluation. Proceedings of the 16th International Workshop on Statistical Modelling, Odense, Denmark, 2--6 July

Jorgensen, B. (1987). Exponential dispersion models. Journal of the Royal Statistical Society, B, 49, 127--162.

Jorgensen, B. (1997). Theory of Dispersion Models. Chapman and Hall, London.

Sidi, Avram (1982). The numerical evaluation of very oscillatory infinite integrals by extrapolation. Mathematics of Computation 38(158), 517--529. 10.1090/S0025-5718-1982-0645667-5

Sidi, Avram (1988). A user-friendly extrapolation method for oscillatory infinite integrals. Mathematics of Computation 51(183), 249--266. 10.1090/S0025-5718-1988-0942153-5

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.

dtweedie.saddle, dtweedie, tweedie.profile, tweedie

##### Aliases
• dtweedie.dldphi
##### Examples
# NOT RUN {
### Plot dl/dphi against candidate values of phi
power <- 2
mu <- 1
phi <- seq(2, 8, by=0.1)

set.seed(10000) # For reproducability
y <- rtweedie( 100, mu=mu, power=power, phi=3)
# So we expect the maximum to occur at  phi=3

dldphi <- dldphi.saddle <- array( dim=length(phi))

for (i in (1:length(phi))) {
dldphi[i] <- dtweedie.dldphi( y=y, power=power, mu=mu, phi=phi[i])
}

plot( dldphi ~ phi, lwd=2, type="l",
ylab=expression(phi), xlab=expression(paste("dl / d",phi) ) )
lines( dldphi.saddle ~ phi, lwd=2, col=2, lty=2)
legend( "bottomright", lwd=c(2,2), lty=c(1,2), col=c(1,2),

# Neither are very good in this case!

# }

Documentation reproduced from package tweedie, version 2.3.2, License: GPL (>= 2)

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