Tweedie {tweedie}

Tweedie Distributions

Description

Density, distribution function, quantile function and random generation for the Tweedie family of distributions

Usage

	dtweedie(y, xi=NULL, mu, phi, power=NULL)
	dtweedie.series(y, power, mu, phi)
	dtweedie.inversion(y, power, mu, phi, exact=TRUE, method)
	dtweedie.stable(y, power, mu, phi)
	ptweedie(q, xi=NULL, mu, phi, power=NULL)
	ptweedie.series(q, power, mu, phi)
	qtweedie(p, xi=NULL, mu, phi, power=NULL)
	rtweedie(n, xi=NULL, mu, phi, power=NULL)

Arguments

y, q

vector of quantiles

p

vector of probabilities

n

the number of observations

xi

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

power

a synonym for \xi

mu

the mean

phi

the dispersion

exact

logical flag; if TRUE (the default), exact zeros are used with the W-algorithm of Sidi (1982); if FALSE, approximate (asymptotic) zeros are used in place of exact zeros. Using asymptotic zeros requires less computation but is often less accurate; using exact zeros can be slower but generally improves accuracy.

method

either 1, 2 or 3, determining which of three methods to use to compute the density using the inversion method. If method is NULL (the default), the optimal method (in terms of relative accuracy) is used, element-by-element of y. See the Note in the Details section below

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.

When 1<p<2, the distribution are continuous for Y greater than zero, with a positive mass at Y=0. For p>2, the distributions are continuous for Y greater than zero.

This function evaluates the density or cumulative probability using one of two methods, depending on the combination of parameters. One method is the evaluation of an infinite series. The second interpolates some stored values computed from a Fourier inversion technique.

The function dtweedie.inversion evaluates the density using a Fourier series technique; ptweedie.inversion does likewise for the cumulative probabilities. The actual code is contained in an external FORTRAN program. Different code is used for p>2 and for 1<p<2.

The function dtweedie.series evaluates the density using a series expansion; a different series expansion is used for p>2 and for 1<p<2. The function ptweedie.series does likewise for the cumulative probabilities but only for 1<p<2.

The function dtweedie.stable exploits the link between the stable distribution (Nolan, 1997) and Tweedie distributions, as discussed in Jorgensen, Chapter 4. These are computed using Nolan's algorithm as implemented in the stabledist package (which is therefore required to use the dtweedie.stable function).

The function dtweedie uses a two-dimensional interpolation procedure to compute the density for some parts of the parameter space from previously computed values found from the series or the inversion. For other parts of the parameter space, the series solution is found.

ptweedie returns either the computed series solution or inversion solution.

Value

density (dtweedie), probability (ptweedie), quantile (qtweedie) or random sample (rtweedie) for the given Tweedie distribution with parameters mu, phi and power.

Note

The methods changed from version 1.4 to 1.5 (methods 1 and 2 swapped). The methods are defined in Dunn and Smyth (2008).

Author(s)

Peter Dunn ([e-mail])

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. doi: 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. doi: 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.

Nolan, John P (1997). Numerical calculation of stable densities and distribution functions. Communication in Statistics—Stochastic models, 13(4). 759–774. doi: 10.1080/15326349708807450

Sidi, Avram (1982). The numerical evaluation of very oscillatory infinite integrals by extrapolation. Mathematics of Computation 38(158), 517–529. doi: 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. doi: 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.

See Also

dtweedie.saddle

Examples

### Plot a Tweedie density
power <- 2.5
mu <- 1 
phi <- 1 
y <- seq(0, 6, length=500) 
fy <- dtweedie( y=y, power=power, mu=mu, phi=phi) 
plot(y, fy, type="l", lwd=2, ylab="Density")
# Compare to the saddlepoint density
f.saddle <- dtweedie.saddle( y=y, power=power, mu=mu, phi=phi) 
lines( y, f.saddle, col=2 )
legend("topright", col=c(1,2), lwd=c(2,1),
    legend=c("Actual","Saddlepoint") )

### A histogram of Tweedie random numbers
hist( rtweedie( 1000, power=1.2, mu=1, phi=1) )

### An example of the multimodal feature of the Tweedie
### family with power near 1 (from Dunn and Smyth, 2005).
y <- seq(0.001,2,len=1000)
mu <- 1
phi <- 0.1
p <- 1.02
f1 <- dtweedie(y,mu=mu,phi=phi,power=p)
plot(y, f1, type="l", xlab="y", ylab="Density")
p <- 1.05
f2<- dtweedie(y,mu=mu,phi=phi,power=p)
lines(y,f2, col=2)

### Compare series and saddlepoint methods
y <- seq(0.001,2,len=1000)
mu <- 1
phi <- 0.1
p <- 1.02
f.series <- dtweedie.series( y,mu=mu,phi=phi,power=p )
f.saddle <- dtweedie.saddle( y,mu=mu,phi=phi,power=p )

f.all <- c( f.series, f.saddle )
plot( range(f.all) ~ range( y ), xlab="y", ylab="Density", 
  type="n")
lines( f.series ~ y, lty=1, col=1)
lines( f.saddle ~ y, lty=3, col=3)

legend("topright", lty=c(1,3), col=c(1,3),
  legend=c("Series","Saddlepoint") )