Package 'tweedie' reference manual

Title:	Evaluation of Tweedie Exponential Family Models
Description:	Maximum likelihood computations for Tweedie families, including the series expansion (Dunn and Smyth, 2005; <doi:10.1007/s11222-005-4070-y>) and the Fourier inversion (Dunn and Smyth, 2008; <doi:10.1007/s11222-007-9039-6>), and related methods.
Authors:	Peter K. Dunn [cre, aut]
Maintainer:	Peter K. Dunn <[email protected]>
License:	GPL (>=2)
Version:	2.3.5
Built:	2024-10-24 05:44:04 UTC
Source:	https://github.com/peterkdunn/tweedie

Tweedie Distributions

Description

Functions for computing and fitting the Tweedie family of distributions

Details

Package:	tweedie
Type:	Package
Version:	2.3.2
Date:	2017-12-14
License:	GPL (>=2)

Author(s)

Peter K Dunn

Maintainer: Peter K Dunn <[email protected]>

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.

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.

Examples

# Generate random numbers
set.seed(987654)
y <- rtweedie( 20, xi=1.5, mu=1, phi=1)
	# With Tweedie index  xi   between 1 and 2, this produces continuous
	# data with exact zeros
x <- rnorm( length(y), 0, 1)  # Unrelated predictor

# With exact zeros, Tweedie index  xi  must be between 1 and 2

# Fit the tweedie distribution; expect  xi  about 1.5
library(statmod)

xi.vec <- seq(1.1, 1.9, by=0.5)
out <- tweedie.profile( y~1, xi.vec=xi.vec, do.plot=TRUE, verbose=TRUE)

# Fit the glm
require(statmod) # Provides  tweedie  family functions
summary(glm( y ~ x, family=tweedie(var.power=out$xi.max, link.power=0) ))
# Generate random numbers
set.seed(987654)
y <- rtweedie( 20, xi=1.5, mu=1, phi=1)
	# With Tweedie index  xi   between 1 and 2, this produces continuous
	# data with exact zeros
x <- rnorm( length(y), 0, 1)  # Unrelated predictor

# With exact zeros, Tweedie index  xi  must be between 1 and 2

# Fit the tweedie distribution; expect  xi  about 1.5
library(statmod)

xi.vec <- seq(1.1, 1.9, by=0.5)
out <- tweedie.profile( y~1, xi.vec=xi.vec, do.plot=TRUE, verbose=TRUE)

# Fit the glm
require(statmod) # Provides  tweedie  family functions
summary(glm( y ~ x, family=tweedie(var.power=out$xi.max, link.power=0) ))

Tweedie Distributions

Description

The AIC for Tweedie glms

Usage

	AICtweedie( glm.obj, dispersion=NULL, k = 2, verbose=TRUE)
AICtweedie( glm.obj, dispersion=NULL, k = 2, verbose=TRUE)

Arguments

`glm.obj`	a fitted Tweedie `glm` object
`dispersion`	the dispersion parameter $\phi$ ; the default is `NULL` which means to use an estimate
`k`	numeric: the penalty per parameter to be used; the default is $k=2$
`verbose`	if `TRUE` (the default), a warning message is produced about the Poisson case; see the second Note below

Details

See AIC for more details on the AIC; see dtweedie for more details on computing the Tweedie densities

Value

Returns a numeric value with the corresponding AIC (or BIC, depending on $k$ )

Note

Computing the AIC may take a long time.

Note

Tweedie distributions with the index parameter as 1 correspond to Poisson distributions when $\phi = 1$ . However, in general a Tweedie distribution with an index parameter equal to one may not be referring to a Poisson distribution with $\phi=1$ , so we cannot assume that $\phi=1$ just because the index parameter is set to one. If the Poisson distribution is intended, then dispersion=1 should be specified. The same argument applies for similar situations.

Author(s)

Peter Dunn ([email protected])

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

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

Sakamoto, Y., Ishiguro, M., and Kitagawa G. (1986). Akaike Information Criterion Statistics. D. Reidel Publishing Company.

Examples

library(statmod) # Needed to use  tweedie  family object

### Generate some fictitious data
test.data <- rgamma(n=200, scale=1, shape=1)

### Fit a Tweedie glm and find the AIC
m1 <- glm( test.data~1, family=tweedie(link.power=0, var.power=2) )

### A Tweedie glm with p=2 is equivalent to a gamma glm:
m2 <- glm( test.data~1, family=Gamma(link=log))

### The models are equivalent, so the AIC shoud be the same:
AICtweedie(m1)
AIC(m2)

library(statmod) # Needed to use  tweedie  family object

### Generate some fictitious data
test.data <- rgamma(n=200, scale=1, shape=1)

### Fit a Tweedie glm and find the AIC
m1 <- glm( test.data~1, family=tweedie(link.power=0, var.power=2) )

### A Tweedie glm with p=2 is equivalent to a gamma glm:
m2 <- glm( test.data~1, family=Gamma(link=log))

### The models are equivalent, so the AIC shoud be the same:
AICtweedie(m1)
AIC(m2)

Tweedie Distributions

Description

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

Usage

dtweedie.dldphi(phi, mu, power, y )
dtweedie.dldphi.saddle(phi, mu, power, y )
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.

Author(s)

Peter Dunn ([email protected])

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.

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

Examples

### 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]) 
   dldphi.saddle[i] <- dtweedie.dldphi.saddle( 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),
   legend=c("'Exact' (using series)","Saddlepoint") )

# Neither are very good in this case!
   
### 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]) 
   dldphi.saddle[i] <- dtweedie.dldphi.saddle( 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),
   legend=c("'Exact' (using series)","Saddlepoint") )

# Neither are very good in this case!

Tweedie Distributions (saddlepoint approximation)

Description

Saddlepoint density for the Tweedie distributions

Usage

dtweedie.saddle(y, xi=NULL, mu, phi, eps=1/6, power=NULL)dtweedie.saddle(y, xi=NULL, mu, phi, eps=1/6, power=NULL)

Arguments

`y`	the vector of responses
`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
`eps`	the offset in computing the variance function. The default is `eps=1/6` (as suggested by Nelder and Pregibon, 1987).

Details

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 approximates the density using the saddlepoint approximation defined by Nelder and Pregibon (1987).

Value

saddlepoint (approximate) density for the given Tweedie distribution with parameters mu, phi and power.

Author(s)

Peter Dunn ([email protected])

References

Daniels, H. E. (1954). Saddlepoint approximations in statistics. Annals of Mathematical Statistics, 25(4), 631–650.

Daniels, H. E. (1980). Exact saddlepoint approximations. Biometrika, 67, 59–63. doi:10.1093/biomet/67.1.59

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 (2001). Tweedie family densities: methods of evaluation. Proceedings of the 16th International Workshop on Statistical Modelling, Odense, Denmark, 2–6 July

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

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.

Nelder, J. A. and Pregibon, D. (1987). An extended quasi-likelihood function. Biometrika, 74(2), 221–232. doi:10.1093/biomet/74.2.221

Examples

p <- 2.5
mu <- 1
phi <- 1
y <- seq(0, 10, length=100)
fy <- dtweedie( y=y, power=p, mu=mu, phi=phi)
plot(y, fy, type="l")
# Compare to the saddlepoint density
f.saddle <- dtweedie.saddle( y=y, power=p, mu=mu, phi=phi)
lines( y, f.saddle, col=2 )
p <- 2.5
mu <- 1
phi <- 1
y <- seq(0, 10, length=100)
fy <- dtweedie( y=y, power=p, mu=mu, phi=phi)
plot(y, fy, type="l")
# Compare to the saddlepoint density
f.saddle <- dtweedie.saddle( y=y, power=p, mu=mu, phi=phi)
lines( y, f.saddle, col=2 )

Tweedie Distributions

Description

The log likelihood for Tweedie models

Usage

	logLiktweedie( glm.obj, dispersion=NULL)
logLiktweedie( glm.obj, dispersion=NULL)

Arguments

`glm.obj`	a fitted Tweedie `glm` object
`dispersion`	the dispersion parameter $\phi$ ; the default is `NULL` which means to use an estimate

Details

The log-likelihood is computed from the AIC, so see AICtweedie for more details.

Value

Returns the log-likelihood from the specified model

Note

Computing the log-likelihood may take a long time.

Note

Author(s)

Peter Dunn ([email protected])

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

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

Sakamoto, Y., Ishiguro, M., and Kitagawa G. (1986). Akaike Information Criterion Statistics. D. Reidel Publishing Company.

Examples

library(statmod) # Needed to use  tweedie  family object

### Generate some fictitious data
test.data <- rgamma(n=200, scale=1, shape=1)

### Fit a Tweedie glm and find the AIC
m1 <- glm( test.data~1, family=tweedie(link.power=0, var.power=2) )

### A Tweedie glm with p=2 is equivalent to a gamma glm:
m2 <- glm( test.data~1, family=Gamma(link=log))

### The models are equivalent, so the AIC shoud be the same:
logLiktweedie(m1)
logLik(m2)

library(statmod) # Needed to use  tweedie  family object

### Generate some fictitious data
test.data <- rgamma(n=200, scale=1, shape=1)

### Fit a Tweedie glm and find the AIC
m1 <- glm( test.data~1, family=tweedie(link.power=0, var.power=2) )

### A Tweedie glm with p=2 is equivalent to a gamma glm:
m2 <- glm( test.data~1, family=Gamma(link=log))

### The models are equivalent, so the AIC shoud be the same:
logLiktweedie(m1)
logLik(m2)

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)
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

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 ([email protected])

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

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") )

### 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") )

Tweedie internal function

Description

Internal tweedie functions.

Usage

	dtweedie.dlogfdphi(y, mu, phi, power)
	dtweedie.logl(phi, y, mu, power)
	dtweedie.logl.saddle( phi, power, y, mu, eps=0)
	dtweedie.logv.bigp( y, phi, power)
	dtweedie.logw.smallp(y, phi, power)
	dtweedie.interp(grid, nx, np, xix.lo, xix.hi,p.lo, p.hi, power, xix)
	dtweedie.jw.smallp(y, phi, power )
	dtweedie.kv.bigp(y, phi, power)
	dtweedie.series.bigp(power, y, mu, phi)
	dtweedie.series.smallp(power, y, mu, phi)
	stored.grids(power)
	twpdf(p, phi, y, mu, exact, verbose, funvalue, exitstatus, relerr, its )
	twcdf(p, phi, y, mu, exact,          funvalue, exitstatus, relerr, its )
dtweedie.dlogfdphi(y, mu, phi, power)
	dtweedie.logl(phi, y, mu, power)
	dtweedie.logl.saddle( phi, power, y, mu, eps=0)
	dtweedie.logv.bigp( y, phi, power)
	dtweedie.logw.smallp(y, phi, power)
	dtweedie.interp(grid, nx, np, xix.lo, xix.hi,p.lo, p.hi, power, xix)
	dtweedie.jw.smallp(y, phi, power )
	dtweedie.kv.bigp(y, phi, power)
	dtweedie.series.bigp(power, y, mu, phi)
	dtweedie.series.smallp(power, y, mu, phi)
	stored.grids(power)
	twpdf(p, phi, y, mu, exact, verbose, funvalue, exitstatus, relerr, its )
	twcdf(p, phi, y, mu, exact,          funvalue, exitstatus, relerr, its )

Arguments

`y`	the vector of responses
`power`	the value of $p$ such that the variance is $\mbox{var}[Y]=\phi\mu^p$
`mu`	the mean
`phi`	the dispersion
`grid`	the interpolation grid necessary for the given value of $p$
`nx`	the number of interpolation points in the $\xi$ dimension
`np`	the number of interpolation points in the $p$ dimension
`xix.lo`	the lower value of the transformed $\xi$ value used in the interpolation grid. (Note that the value of $\xi$ is from $0$ to $\infty$ , and is transformed such that it is on the range $0$ to $1$ .)
`xix.hi`	the higher value of the transformed $\xi$ value used in the interpolation grid.
`p.lo`	the lower value of $p$ value used in the interpolation grid.
`p.hi`	the higher value of $p$ value used in the interpolation grid.
`xix`	the value of the transformed $\xi$ at which a value is sought.
`eps`	the offset in computing the variance function in the saddlepoint approximation. The default is `eps=1/6` (as suggested by Nelder and Pregibon, 1987).
`p`	the Tweedie index parameter
`exact`	a flag for the FORTRAN to use exact-zeros acceleration algorithmic the calculation (1 means to do so)
`verbose`	a flag for the FORTRAN: 1 means to be verbose
`funvalue`	the value of the call returned by the FORTRAN code
`exitstatus`	the exit status returned by the FORTRAN code
`relerr`	an estimation of the relative error returned by the FORTRAN code
`its`	the number of iterations of the algorithm returned by the FORTRAN code

Details

These are not to be called by the user.

Author(s)

Peter Dunn ([email protected])

References

Nelder, J. A. and Pregibon, D. (1987). An extended quasi-likelihood function Biometrika, 74(2), 221–232. doi10.1093/biomet/74.2.221

Convert Tweedie parameters

Description

Converts Tweedie distribution parameters to the parameters of the underlying distributions

Usage

	tweedie.convert( xi=NULL, mu, phi, power=NULL)
tweedie.convert( xi=NULL, mu, phi, power=NULL)

Arguments

`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

Details

The Tweedie family of distributions with $1<\xi<2$ is the Poisson sum of gamma distributions (where the Poisson distribution has mean $\lambda$ , and the gamma distribution has scale and shape parameters). When used to fit a glm, the model is fitted with the usual glm parameters: the mean $\mu$ and the dispersion parameter $\phi$ . This function converts the parameters $(p, \mu, \phi)$ to the values of the parameters of the underlying Poisson distribution $\lambda$ and gamma distribution (scale and shape parameters).

Value

a list containing the values of the mean of the underlying Poisson distribution (as poisson.lambda), the scale parameter of the underlying gamma distribution (as gamma.scale), the shape parameter of the underlying gamma distribution (as gamma.shape), the probability of obtaining a zero response (as p0), the mean of the underlying gamma distribution (as gamma.mean), and the dispersion parameter of the underlying gamma distribution (as gamma.phi).

Author(s)

Peter Dunn ([email protected])

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

Examples

tweedie.convert(xi=1.5, mu=1, phi=1)
tweedie.convert(xi=1.5, mu=1, phi=1)

Tweedie Distributions: the deviance function

Description

The deviance function for the Tweedie family of distributions

Usage

tweedie.dev(y, mu, power)
tweedie.dev(y, mu, power)

Arguments

`y`	vector of quantiles (which can be zero if $1<p<2$
`mu`	the mean
`power`	the value of $p$ such that the variance is $\mbox{var}[Y]=\phi\mu^p$

Details

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 ([email protected])

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.

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

Examples

### 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") )

### 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") )

Tweedie Distributions: plotting

Description

Plotting Tweedie density and distribution functions

Usage

   tweedie.plot(y, xi, mu, phi, type="pdf", power=NULL, add=FALSE, ...)
tweedie.plot(y, xi, mu, phi, type="pdf", power=NULL, add=FALSE, ...)

Arguments

`y`	vector of values at which to evaluate and plot
`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
`type`	what to plot: `pdf` (the default) means the probability function, or `cdf`, the cumulative distribution function
`add`	if `TRUE`, the plot is added to the current device; if `FALSE` (the default), a new plot is produced
`...`	Arguments to be passed to the plotting method

Details

For details, see dtweedie

Value

this function is usually called for side-effect of producing a plot of the specified Tweedie distribution, properly plotting the exact zero that occurs at $y=0$ when $1<p<2$ . However, it also produces a list with the computed density at the given points, with components y and x respectively, such that plot(y~x) approximately reproduces the plot.