Version: 2.3.5
Date: 2022-08-17
Title: Evaluation of Tweedie Exponential Family Models
Author: Peter K. Dunn [cre, aut]
Maintainer: Peter K. Dunn <pdunn2@usc.edu.au>
Depends: R (≥ 2.8.0)
Imports: methods
Suggests: stabledist, statmod(≥ 1.4.20)
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.
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
NeedsCompilation: yes
RoxygenNote: 6.0.1
Packaged: 2022-08-17 01:02:19 UTC; pdunn2
Repository: CRAN
Date/Publication: 2022-08-17 07:40:02 UTC

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 <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. 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) ))

Tweedie Distributions

Description

The AIC for Tweedie glms

Usage

	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 (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. 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.

See Also

AIC

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)

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 (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. 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") )

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 )

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 (pdunn2@usc.edu.au)

References

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

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 )

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 (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. 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

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, dtweedie, tweedie.profile, tweedie

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!

Tweedie Distributions (saddlepoint approximation)

Description

Saddlepoint density for the Tweedie distributions

Usage

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

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 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 (pdunn2@usc.edu.au)

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

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

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 )

Tweedie Distributions

Description

The log likelihood for Tweedie models

Usage

	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

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 (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. 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.

See Also

AICtweedie

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)

Convert Tweedie parameters

Description

Converts Tweedie distribution parameters to the parameters of the underlying distributions

Usage

	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 (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. 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

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

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)

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 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.

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. 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

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

### 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, ...)

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.

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. 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

Examples

### Plot a Tweedie density with 1<p<2
yy <- seq(0,5,length=100)
tweedie.plot( power=1.7, mu=1, phi=1, y=yy, lwd=2)
tweedie.plot( power=1.2, mu=1, phi=1, y=yy, add=TRUE, lwd=2, col="red")
legend("topright",lwd=c(2,2), col=c("black","red"), pch=c(19,19),
   legend=c("p=1.7","p=1.2") )

### Plot distribution functions
tweedie.plot( power=1.05, mu=1, phi=1, y=yy,
   lwd=2, type="cdf", ylim=c(0,1))
tweedie.plot( power=2, mu=1, phi=1, y=yy, 
   add=TRUE, lwd=2, type="cdf",col="red")
legend("bottomright",lwd=c(2,2), col=c("black","red"),
   legend=c("p=1.05","p=2") )

### Now, plot two densities, combining p>2 and 1<p<2
tweedie.plot( power=3.5, mu=1, phi=1, y=yy, lwd=2)
tweedie.plot( power=1.5, mu=1, phi=1, y=yy, lwd=2, col="red", add=TRUE)
legend("topright",lwd=c(2,2), col=c("black","red"), pch=c(NA,19),
   legend=c("p=3.5","p=1.5") )

Tweedie Distributions: mle estimation of p

Description

Maximum likelihood estimation of the Tweedie index parameter p.

Usage

tweedie.profile(formula, p.vec=NULL, xi.vec=NULL, link.power=0, 
      data, weights, offset, fit.glm=FALSE,
      do.smooth=TRUE, do.plot=FALSE, do.ci=do.smooth,
      eps=1/6, 
      control=list( epsilon=1e-09, maxit=glm.control()$maxit, trace=glm.control()$trace ),
      do.points=do.plot, method="inversion", conf.level=0.95, 
      phi.method=ifelse(method == "saddlepoint", "saddlepoint", "mle"), 
      verbose=FALSE, add0=FALSE)

Arguments

formula

a formula expression as for other regression models and generalized linear models, of the form response ~ predictors. For details, see the documentation for lm, glm and formula

p.vec

a vector of p values for consideration. The values must all be larger than one (if the response variable has exact zeros, the values must all be between one and two). If NULL (the default), p.vec is set to seq(1.2, 1.8, by=0.1) if the response contains any zeros, or seq(1.5, 5, by=0.5) if the response contains no zeros. See the DETAILS section below for further details.

xi.vec

the same as p.vec; some authors use the p notation for the index parameter, and some use \xi; this function detects which is used and then uses that notation throughout

link.power

the power link function to use. These link functions g(\cdot) are of the form g(\eta)=\eta^{\rm link.power}, and the special case of link.power=0 (the default) refers to the logarithm link function. See the documentation for tweedie also.

data

an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which glm is called.

weights

an optional vector of weights to be used in the fitting process. Should be NULL or a numeric vector.

offset

this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length either one or equal to the number of cases. One or more offset terms can be included in the formula instead or as well, and if both are specified their sum is used. See model.offset.

fit.glm

logical flag. If TRUE, the Tweedie generalized linear model is fitted using the value of p found by the profiling function. If FALSE (the default), no model is fitted.

do.smooth

logical flag. If TRUE (the default), a spline is fitted to the data to smooth the profile likelihood plot. If FALSE, no smoothing is used (and the function is quicker). Note that p.vec must contain at least five points for smoothing to be allowed.

do.plot

logical flag. If TRUE, a plot of the profile likelihood is produce. If FALSE (the default), no plot is produced.

do.ci

logical flag. If TRUE, the nominal 100*conf.level is computed. If FALSE, no confidence interval is computed. By default, do.ci is the same value as do.smooth, since a confidence interval will only be accurate if smoothing has been performed. Indeed, if do.smooth=FALSE, confidence intervals are never computed and do.ci is forced to FALSE if it is given as TRUE.

eps

the offset in computing the variance function. The default is eps=1/6 (as suggested by Nelder and Pregibon, 1987). Note eps is ignored unless the method="saddlepoint" as it makes no sense otherwise.

control

a list of parameters for controlling the fitting process; see glm.control and glm. The default is to use the maximum number of iterations maxit and the trace setting as given in glm.control, but to set epsilon to 1e-09 to ensure a smoother plot

do.points

plot the points on the plot where the (log-) likelihood is computed for the given values of p; defaults to the same value as do.plot

method

the method for computing the (log-) likelihood. One of "series", "inversion" (the default), "interpolation" or "saddlepoint". If there are any troubles using this function, sometimes a change of method will fix the problem. Note that method="saddlepoint" is only an approximate method for computing the (log-) likelihood. Using method="interpolation" may produce a jump in the profile likelihood as it changes computational regimes.

conf.level

the confidence level for the computation of the nominal confidence interval. The default is conf.level=0.95.

phi.method

the method for estimating phi, one of "saddlepoint" or "mle". A maximum likelihood estimate is used unless method="saddlepoint", when the saddlepoint approximation method is used. Note that using phi.method="saddlepoint" is equivalent to using the mean deviance estimator of phi.

verbose

the amount of feedback requested: 0 or FALSE means minimal feedback (the default), 1 or TRUE means some feedback, or 2 means to show all feedback. Since the function can be slow and sometimes problematic, feedback can be good; but it can also be unnecessary when one knows all is well.

add0

if TRUE, the value p=0 is used in forming the profile log-likelihood (corresponding to the normal distribution); the default value is add0=FALSE

Details

For each value in p.vec, the function computes an estimate of phi and then computes the value of the log-likelihood for these parameters. The plot of the log-likelihood against p.vec allows the maximum likelihood value of p to be found. Once the value of p is found, the distribution within the class of Tweedie distribution is identified.

Value

The main purpose of the function is to estimate the value of the Tweedie index parameter, p, which is produced by the output list as p.max. Optionally (if do.plot=TRUE), a plot is produced that shows the profile log-likelihood computed at each value in p.vec (smoothed if do.smooth=TRUE). This function can be temperamental (for theoretical reasons involved in numerically computing the density), and this plot shows the values of p requested on the horizontal axis (using rug); there may be fewer points on the plot, since the likelihood some values of p requested may have returned NaN, Inf or NA.

A list containing the components: y and x (such that plot(x,y) (partially) recreates the profile likelihood plot); ht (the height of the nominal confidence interval); L (the estimate of the (log-) likelihood at each given value of p); p (the p-values used); phi (the computed values of phi at the values in p); p.max (the estimate of the mle of p); L.max (the estimate of the (log-) likelihood at p.max); phi.max (the estimate of phi at p.max); ci (the lower and upper limits of the confidence interval for p); method (the method used for estimation: series, inversion, interpolation or saddlepoint); phi.method (the method used for estimation of phi: saddlepoint or phi).

If glm.fit is TRUE, the list also contains a component glm.obj, a glm object for the fitted Tweedie generalized linear model.

Note

The estimates of p and phi are printed. The result is printed invisibly.

If the response variable has any exact zeros, the values in p.vec must all be between one and two.

The function is sometimes unstable and may fail. It may also be very slow. One solution is to change the method. The default is method="inversion" (the default); then try method="series", method="interpolation" and method="saddlepoint" in that order. Note that method="saddlepoint" is an approximate method only. Also make sure the values in p.vec are suitable for the data (see above paragraph).

It is recommended that for the first use with a data set, use p.vec with only a small number of values and set do.smooth=FALSE, do.ci=FALSE. If this is successful, a larger vector p.vec and smoothing can be used.

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. 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.

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

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

Examples

library(statmod) # Needed to use  tweedie.profile
# Generate some fictitious data
test.data <- rgamma(n=200, scale=1, shape=1)
# The gamma is a Tweedie distribution with power=2;
# let's see if p=2 is suggested by  tweedie.profile:
## Not run: 
	out <- tweedie.profile( test.data ~ 1, 
		p.vec=seq(1.5, 2.5, by=0.2) )
	out$p.max
	out$ci

## End(Not run)