Tweedie Distributions: the deviance function

Description

The deviance function for the Tweedie family of distributions

Usage

1

Arguments

y

vector of quantiles (which can be zero if 1 < power < 2

mu

the mean

power

the value of power such that the variance is var(Y) = phi * mu^power

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 var(Y) = phi * mu^power where power is greater than or equal to one, or less than or equal to zero. This function only evaluates for power greater than or equal to one. Special cases include the normal (power=0), Poisson (power=1 with phi=1), gamma (power=2) and inverse Gaussian (power=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.

The deviance is defined by deviance as “up to a constant, minus twice the maximized log-likelihood. Where sensible, the constant is chosen so that a saturated model has deviance zero.”

Value

the value of the deviance for the given Tweedie distribution with parameters mu, phi and power.

Author(s)

Peter Dunn (pdunn2@usc.edu.au)

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.

Dunn, Peter K and Smyth, Gordon K (2005). Series evaluation of Tweedie exponential dispersion model densities Statistics and Computing, 15(4). 267–280.

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.

Sidi, Avram (1988). A user-friendly extrapolation method for oscillatory infinite integrals. Mathematics of Computation 51(183), 249–266.

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.

See Also

dtweedie, dtweedie.saddle, tweedie, deviance, glm

Examples

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### Plot a Tweedie deviance function when 1<p<2
mu <- 1 

y <- seq(0, 6, length=100) 

dev1 <- tweedie.dev( y=y, mu=mu, power=1.1) 
dev2 <- tweedie.dev( y=y, mu=mu, power=1.5)
dev3 <- tweedie.dev( y=y, mu=mu, power=1.9) 

plot(range(y), range( c(dev1, dev2, dev3)), 
   type="n", lwd=2, ylab="Deviance", xlab=expression(italic(y)) )

lines( y, dev1, lty=1, col=1, lwd=2 )
lines( y, dev2, lty=2, col=2, lwd=2 )
lines( y, dev3, lty=3, col=3, lwd=2 )


legend("top", col=c(1,2,3), lwd=c(2,2,2), lty=c(1,2,3),
    legend=c("p=1.1","p=1.5", "p=1.9") )


### Plot a Tweedie deviance function when p>2
mu <- 1 

y <- seq(0.1, 6, length=100) 

dev1 <- tweedie.dev( y=y, mu=mu, power=2) # Gamma
dev2 <- tweedie.dev( y=y, mu=mu, power=3) # Inverse Gaussian
dev3 <- tweedie.dev( y=y, mu=mu, power=4) 

plot(range(y), range( c(dev1, dev2, dev3)), 
   type="n", lwd=2, ylab="Deviance", xlab=expression(italic(y)) )

lines( y, dev1, lty=1, col=1, lwd=2 )
lines( y, dev2, lty=2, col=2, lwd=2 )
lines( y, dev3, lty=3, col=3, lwd=2 )


legend("top", col=c(1,2,3), lwd=c(2,2,2), lty=c(1,2,3),
    legend=c("p=2 (gamma)", "p=3 (inverse Gaussian)", "p=4") )