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:	3.1.0
Built:	2026-05-24 23:08:58 UTC
Source:	https://github.com/peterkdunn/tweedie

Evaluation of Tweedie Exponential Family Models

Description

This package provides maximum likelihood computations for Tweedie families, including the series expansion (Dunn and Smyth, 2005) and the Fourier inversion (Dunn and Smyth, 2008), and related methods.

Author(s)

Maintainer: Peter K. Dunn [email protected]

Fourier Inversion Evaluation for the Tweedie Probability Function

Description

Evaluates the probability density function (pdf) for Tweedie distributions using Fourier inversion, for given values of the dependent variable y, the mean mu, dispersion phi, and power parameter power. Not usually called by general users, but can be used in the case of evaluation problems.

Usage

dtweedie_inversion(y, mu, phi, power, method = 3, verbose = FALSE, 
                          details = FALSE, IGexact = TRUE)

dtweedie.inversion(y, power, mu, phi, method = 3, verbose, details)
dtweedie_inversion(y, mu, phi, power, method = 3, verbose = FALSE, 
                          details = FALSE, IGexact = TRUE)

dtweedie.inversion(y, power, mu, phi, method = 3, verbose, details)

Arguments

y

vector of quantiles.

mu

the mean parameter $\mu$ .

phi

the dispersion parameter $\phi$ .

power

scalar; the power parameter $p$ .

method

the method to use; one of 1, 2, or 3 (the default).

verbose

logical; if TRUE, display some internal computation details. The default is FALSE.

details

logical; if TRUE, return a list with basic details of the integration. The default is FALSE.

IGexact

logical; if TRUE (the default), evaluate the inverse Gaussian distribution using the 'exact' values, otherwise uses inversion.

Value

A numeric vector of densities if details=FALSE; if details = TRUE, a list containing denisty (a vector of the values of the density), regions (a vector of the number of integration regions used),method (a vector giving the evaluation method used; see the Note below on the three methods), and exitstatus (a vector, where a 1 for any value means a computational problem or target relative accuracy not reached, for the corresponding observation).

Note

The 'exact' values for the inverse Gaussian distribution are not really exact, but evaluated using inverse normal distributions, for which very good numerical approximation are available in R. For special cases of $p$ (i.e., $p = 0, 1, 2, 3$ ), where no inversion is needed, regions and method are set to NA for all values of y. For special cases of y for other values of $p$ (i.e., $P(Y = 0)$ ), regions and method are set to NA.

The three methods are described in Dunn & Smyth (2008).

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

Examples

# Plot a Tweedie density
y <- seq(0.02, 4, length = 50)
fy <- dtweedie_inversion(y, mu = 1, phi = 1, power = 1.1)
plot(y, fy, type = "l", lwd = 2, ylab = "Density")

# Plot a Tweedie density
y <- seq(0.02, 4, length = 50)
fy <- dtweedie_inversion(y, mu = 1, phi = 1, power = 1.1)
plot(y, fy, type = "l", lwd = 2, ylab = "Density")

Tweedie densities evaluation using the saddlepoint approximation

Description

Density function for the Tweedie EMDs using a saddlepoint approximation.

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)
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

vector of quantiles.

xi

scalar; the value of $\xi$ such that the variance is $\mbox{var}[Y]=\phi\mu^{\xi}$ . A synonym for power.

mu

vector of mean $\mu$ .

phi

vector of dispersion parameters $\phi$ .

eps

the offset in computing the variance function; the default is eps=1/6 (as suggested by Nelder and Pregibon, 1987).

power

scalar; a synonym for $\xi$ , the Tweedie index parameter.

Value

A numeric vector of densities.

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

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

# Plot a Tweedie density
y <- seq(0.01, 4, length = 50)
fy <- dtweedie_saddle(y, power = 1.1, mu = 1, phi = 1)
plot(y, fy, type = "l", lwd = 2, ylab = "Density")


# Plot a Tweedie density
y <- seq(0.01, 4, length = 50)
fy <- dtweedie_saddle(y, power = 1.1, mu = 1, phi = 1)
plot(y, fy, type = "l", lwd = 2, ylab = "Density")

Series Evaluation for the Tweedie Probability Function

Description

Evaluates the probability density function (pdf) for Tweedie distributions using an infinite series, for given values of the dependent variable y, the mean mu, dispersion phi, and power parameter power. Not usually called by general users, but can be used in the case of evaluation problems.

Usage

dtweedie_series(y, power, mu, phi, details = FALSE)

dtweedie.series(y, power, mu, phi)
dtweedie_series(y, power, mu, phi, details = FALSE)

dtweedie.series(y, power, mu, phi)

Arguments

y

vector of quantiles.

power

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

mu

vector of mean $\mu$ .

phi

vector of dispersion parameters $\phi$ .

details

logical; if TRUE, returns the value of the distribution function and some details.

Value

A numeric vector of densities.

References

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

Examples

# Plot a Tweedie density
y <- seq(0.01, 4, length = 50)
fy <- dtweedie_series(y, power = 1.1, mu = 1, phi = 1)
plot(y, fy, type = "l", lwd = 2, ylab = "Density")
# Plot a Tweedie density
y <- seq(0.01, 4, length = 50)
fy <- dtweedie_series(y, power = 1.1, mu = 1, phi = 1)
plot(y, fy, type = "l", lwd = 2, ylab = "Density")

Log-likelihood for Tweedie distributions

Description

Evaluates the log-likelihood for a fitted Tweedie glm.

Usage

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

Arguments

glm.obj

a fitted glm object, fitted using the tweedie family.

dispersion

the dispersion parameter, usually extracted from glm.obj; however, occasionally a specified value of the dispersion may be needed.

Details

The log-Likelihood is computed by evaluating the density function.

Value

The value of the computed log-likelihood.

Note

Evaluating the likelihood can be time consuming, so the function may take some time for large data sets.

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

# Fit a Tweedie density using  tweedie  family function from  statmod
pretend <- data.frame( y = stats::rgamma(20, shape = 1, rate = 1) )
fit <- glm(y ~ 1, data = pretend, 
           family = statmod::tweedie(link.power = 0, var.power = 2.1))

# Compute the AIC
logLiktweedie(fit)

# Fit a Tweedie density using  tweedie  family function from  statmod
pretend <- data.frame( y = stats::rgamma(20, shape = 1, rate = 1) )
fit <- glm(y ~ 1, data = pretend, 
           family = statmod::tweedie(link.power = 0, var.power = 2.1))

# Compute the AIC
logLiktweedie(fit)

Fourier Inversion Evaluation for the Tweedie Distribution Function

Description

Evaluates the distribution function (df) for Tweedie distributions using Fourier inversion, for given values of the dependent variable y, the mean mu, dispersion phi, and power parameter power. Not usually called by general users, but can be in the case of evaluation problems.

Usage

ptweedie_inversion(q, mu, phi, power, verbose = FALSE, details = FALSE, IGexact = TRUE)

ptweedie.inversion(q, power, mu, phi, verbose, details)
ptweedie_inversion(q, mu, phi, power, verbose = FALSE, details = FALSE, IGexact = TRUE)

ptweedie.inversion(q, power, mu, phi, verbose, details)

Arguments

q

vector of quantiles.

mu

the mean parameter.

phi

the dispersion parameter.

power

the power parameter $p$ .

verbose

logical; if TRUE, displays some internal computation details. The default is FALSE.

details

logical; if TRUE, returns the value of the distribution and some information about the integration. The default is FALSE.

IGexact

logical; if TRUE (the default), evaluate the inverse Gaussian distribution using the 'exact' values, otherwise uses inversion.

Value

If details = FALSE, a numeric vector of the distribution function values; if details = TRUE, a list containing CDF (a vector of the values of the distribution function), regions (a vector of the number of integration regions used), and exitstatus (a vector, where a 1 for any value means a computational problem or target relative accuracy not reached, for the corresponding observation).

For special cases of $p$ (i.e., $p = 0, 1, 2, 3$ ), where no inversion is needed, regions is set to NA for all values of q. For special cases of q for other values of $p$ (i.e., $P(Y = 0)$ ), regions is set to NA.

Note

The 'exact' values for the inverse Gaussian distribution are not really exact, but evaluated using inverse normal distributions, for which very good numerical approximation are available in R.

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

Examples

# Plot a Tweedie distribution function
y <- seq(0.01, 4, length = 50)
Fy <- ptweedie_inversion(y, mu = 1, phi = 1, power = 1.1)
plot(y, Fy, type = "l", lwd = 2, ylab = "Distribution function")

# Plot a Tweedie distribution function
y <- seq(0.01, 4, length = 50)
Fy <- ptweedie_inversion(y, mu = 1, phi = 1, power = 1.1)
plot(y, Fy, type = "l", lwd = 2, ylab = "Distribution function")

Series Evaluation for the Tweedie Distribution Function

Description

Evaluates the distribution function (df) for Tweedie distributions with $1 < p < 2$ using an infinite series, for given values of the dependent variable y, the mean mu, dispersion phi, and power parameter power. Not usually called by general users, but can be in the case of evaluation problems.

Usage

ptweedie_series(q, power, mu, phi, verbose = FALSE, details = FALSE)

ptweedie.series(q, power, mu, phi, verbose = FALSE, details = FALSE)
ptweedie_series(q, power, mu, phi, verbose = FALSE, details = FALSE)

ptweedie.series(q, power, mu, phi, verbose = FALSE, details = FALSE)

Arguments

q

vector of quantiles.

power

the power parameter $p$ .

mu

the mean parameter $\mu$ .

phi

the dispersion parameter $\phi$ .

verbose

logical; if TRUE, displays some internal computation details. The default is FALSE.

details

logical; if TRUE, returns the value of the distribution function and some details.

Value

A numeric vector of densities.

Note

The 'exact' values for the inverse Gaussian distribution are not really exact, but evaluated using inverse normal distributions, for which very good numerical approximation are available in R.

References

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

Examples

# Plot a Tweedie distribution function
y <- seq(0.01, 4, length = 50)
Fy <- ptweedie_series(y, power = 1.1, mu = 1, phi = 1)
plot(y, Fy, type = "l", lwd = 2, ylab = "Distribution function")

# Plot a Tweedie distribution function
y <- seq(0.01, 4, length = 50)
Fy <- ptweedie_series(y, power = 1.1, mu = 1, phi = 1)
plot(y, Fy, type = "l", lwd = 2, ylab = "Distribution function")

Tweedie distributions

Description

Density, distribution function, quantile function and random generation for the the Tweedie family of distributions, with mean mu, dispersion parameter phi and variance power power (or xi, a synonym for power).

Usage

dtweedie(y, xi = NULL, mu, phi, power = NULL, verbose = FALSE)

ptweedie(q, xi = NULL, mu, phi, power = NULL, verbose = FALSE)

qtweedie(p, xi = NULL, mu, phi, power = NULL)

rtweedie(n, xi = NULL, mu, phi, power = NULL)

ptweedie(q, xi = NULL, mu, phi, power = NULL, verbose = FALSE)

qtweedie(p, xi = NULL, mu, phi, power = NULL)

rtweedie(n, xi = NULL, mu, phi, power = NULL)
dtweedie(y, xi = NULL, mu, phi, power = NULL, verbose = FALSE)

ptweedie(q, xi = NULL, mu, phi, power = NULL, verbose = FALSE)

qtweedie(p, xi = NULL, mu, phi, power = NULL)

rtweedie(n, xi = NULL, mu, phi, power = NULL)

ptweedie(q, xi = NULL, mu, phi, power = NULL, verbose = FALSE)

qtweedie(p, xi = NULL, mu, phi, power = NULL)

rtweedie(n, xi = NULL, mu, phi, power = NULL)

Arguments

y

vector of quantiles.

xi

scalar; the value of $\xi$ such that the variance is $\mbox{var}[Y]=\phi\mu^{\xi}$ . A synonym for power.

mu

vector of mean $\mu$ .

phi

vector of dispersion parameters $\phi$ .

power

scalar; a synonym for $\xi$ , the Tweedie index parameter.

verbose

logical; if TRUE, some details of the algorithms used is shown. The default is FALSE.

q

vector of quantiles.

p

vector of probabilities.

n

number of observations.

Details

The Tweedie edms belong to the class of exponential dispersion models (edms), known for their role in generalized linear models (glms). The Tweedie distributions are the edms with a variance of the form $\mbox{var}[Y] = \phi\mu^p$ where $p \ge 1$ . This function only evaluates for $p \ge 1$ .

Special cases are the Poisson ( $p = 1$ with $\phi = 1$ ), gamma ( $p = 2$ ), and inverse Gaussian ( $p = 3$ ) distributions. Evaluation is difficult for $p$ outside of $p = 0, 1, 2, 3$ . This function uses one of two primary methods, depending on the combination of parameters:

Evaluation of an infinite series (dtweedie_series).
Interpolation from stored values computed via a Fourier inversion technique (dtweedie_inversion).

This function employs a two-dimensional interpolation procedure to compute the density for some parts of the parameter space from previously computed values (interpolation) and uses the series solution for others.

When $1<p<2$ , the density function include a positive probably for $Y = 0$ .

Value

dtweedie gives the density, ptweedie gives the distribution function, qtweedie gives the quantile function, and rtweedie generates random deviates.

The length of the result is determined by n for rtweedie, and by the length of mu for other functions.

Note

dtweedie and ptweedie are the only functions generally to be called by users. Consequently, all checks on the function inputs are performed in these functions.

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.

Examples

# Compute a Tweedie density
power <- 1.1
mu <- 1
phi <- 1
y <- seq(0, 5, by = 0.5)
dtweedie(y, power = power, mu = mu, phi = phi)

# Compare to the saddlepoint density
dtweedie_saddle(y = y, power = power, mu = mu, phi = phi)

# The DF:
ptweedie(y, power = power, mu = mu, phi = phi)

# Compute a Tweedie density
power <- 1.1
mu <- 1
phi <- 1
y <- seq(0, 5, by = 0.5)
dtweedie(y, power = power, mu = mu, phi = phi)

# Compare to the saddlepoint density
dtweedie_saddle(y = y, power = power, mu = mu, phi = phi)

# The DF:
ptweedie(y, power = power, mu = mu, phi = phi)

AIC for Tweedie Glms

Description

Evaluates the aic for a fitted Tweedie glm. 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 $p \ge 1$ . This function only evaluates for $p \ge 1$ .

Usage

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

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

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

Arguments

glm.obj

a fitted glm object, fitted using the tweedie family.

dispersion

the dispersion parameter, usually extracted from glm.obj; however, occasionally a specified value of the dispersion may be needed.

k

the aic penalty; k = 2 (the default) produces the AIC.

verbose

logical; if TRUE, display details of the internal process. The default is FALSE.

Details

The aic is computed by evaluating the density function.

Value

The value of the computed aic.

Note

Evaluating the likelihood can be time consuming, so the function may take some time for large data sets.

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

# Fit a Tweedie density using  tweedie  family function from  statmod
pretend <- data.frame( y = stats::rgamma(20, shape = 1, rate = 1) )
fit <- glm(y ~ 1, data = pretend, 
           family = statmod::tweedie(link.power = 0, var.power = 2.1))

# Compute the AIC
tweedie_AIC(fit)

# Fit a Tweedie density using  tweedie  family function from  statmod
pretend <- data.frame( y = stats::rgamma(20, shape = 1, rate = 1) )
fit <- glm(y ~ 1, data = pretend, 
           family = statmod::tweedie(link.power = 0, var.power = 2.1))

# Compute the AIC
tweedie_AIC(fit)

Tweedie Distribution: Convert Between Parameter Formats

Description

Converts from the fitted glm parameters $p$ , $\mu$ and $\phi$ and the corresponding underlying Poisson and gamma parameters (when $1 < p < 2$ ).

Usage

tweedie_convert(xi = NULL, mu, phi, power = NULL)

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

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

Arguments

xi

a synonym for power.

mu

the mean parameter $\mu$ .

phi

the dispersion parameter $\phi$ .

power

the power parameter $p$ ; a synonym for $\xi$ .

Value

a list of the parameters of the parameters of the corresponding underlying Poisson and gamma densities: poisson.lambda ( $\lambda$ from the underlying Poisson distribution), gamma.shape, gamma.scale (the shape and scale parameters from the underlying gamma distribution), p0 (the probability that $Y = 0$ ), gamma.mean and gamma.phi (the gamma mean and dispersion parameter values)

Examples

### Fit a Tweedie density
pretend <- data.frame( y = rgamma(20, shape = 1, rate = 1) )
fit <- glm(y ~ 1, data = pretend, 
           family = statmod::tweedie(link.power = 0, var.power = 1.4))

# Convert parameters
tweedie_convert(mu = fitted(fit, type="response"), phi = 1, power = 1.4)

### Fit a Tweedie density
pretend <- data.frame( y = rgamma(20, shape = 1, rate = 1) )
fit <- glm(y ~ 1, data = pretend, 
           family = statmod::tweedie(link.power = 0, var.power = 1.4))

# Convert parameters
tweedie_convert(mu = fitted(fit, type="response"), phi = 1, power = 1.4)

Unit Deviance for a Tweedie Distribution

Description

Computes the unit deviance for Tweedie distributions.

Usage

tweedie_dev(y, mu, power)

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

tweedie.dev(y, mu, power)

Arguments

y

vector of quantiles.

mu

the mean parameter $\mu$ .

power

the power parameter $p$ .

Value

A numeric vector containing the unit deviance.

References

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

Examples

# Unit deviance is not symmetric in general:
round( tweedie_dev(0:6, mu = 3, power = 1.1), 3)

# Unit deviance is not symmetric in general:
round( tweedie_dev(0:6, mu = 3, power = 1.1), 3)

Display Integrand Information for Tweedie Fourier inversion

Description

Plots the integrand for Fourier inversion and the real and imaginary parts separately.

Usage

tweedie_integrand(y, power, mu, phi, t = seq(0, 5, length = 200), 
                  type = "PDF", whichPlots = 1:4, 
                  plot_args  = list())
tweedie_integrand(y, power, mu, phi, t = seq(0, 5, length = 200), 
                  type = "PDF", whichPlots = 1:4, 
                  plot_args  = list())

Arguments

y

vector of quantiles.

power

a synonym for $\xi$ ; the Tweedie power-index on the variance.

mu

the mean parameter $\mu$ .

phi

the dispersion parameter $\phi$ .

t

the values of the variable over which to integrate; the default is t = seq(0, 5, length = 200).

type

either "PDF" (the default) for the (probability) density function, or "CDF" for the (cumulative) distribution function.

whichPlots

which combination of the four plots (described below) are produced; by default, all four are produced (i.e., whichPlots = 1:4).

plot_args

A named list of arguments controlling the main plot. These are passed to plot.

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 displays the integrand that is evaluated for computing the Fourier inversion, for the PDF or CDF.

Value

A list containing the real and imaginary parts of $k(t)$ , Real and Imag respectively, plus the values of the integrand as IG. The main purpose of the function is the side-effect of producing a $2\times2$ grid of plots. The first is the imaginary parts of $k(t)$ . The second is $\cos\Im k(t)$ (for type=="PDF") or $\sin\Im k(t)$ (for type=="CDF"). The third is the real part of $\Re k(t)$ The fourth is the integrand, with the envelope shown as a dashed line.

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.

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

tweedie_integrand(2, power = 3, mu = 1, phi = 1,
                  plot_args = list(lwd = 2))

tweedie_integrand(2, power = 3, mu = 1, phi = 1,
                  plot_args = list(lwd = 2))

The Probability of Observing a Zero Value for a Tweedie Density

Description

The probability that the variable takes the value of zero.

Usage

tweedie_lambda(mu, phi, power)
tweedie_lambda(mu, phi, power)

Arguments

mu

the mean parameter $\mu$ .

phi

the dispersion parameter $\phi$ .

power

the power parameter $p$ (sometimes denoted $\xi$ ).

Value

The value of $\lambda$ when $1 < p < 2$ such that $P(Y=0) = \exp(-\lambda)$ . When $p>2$ , a vector of zeros is returned.

References

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

Examples

lambda <- tweedie_lambda(mu = 1:3, phi = 1, power = 1.1)
exp( -lambda)

# When p > 2, there is zero probability that Y = 0:
lambda  <- tweedie_lambda(mu = 1, phi = 1, power = 3.1)

lambda <- tweedie_lambda(mu = 1:3, phi = 1, power = 1.1)
exp( -lambda)

# When p > 2, there is zero probability that Y = 0:
lambda  <- tweedie_lambda(mu = 1, phi = 1, power = 3.1)

Plot Tweedie Models

Description

This function produced a plot of the specified Tweedie distribution.

Usage

tweedie_plot(y, xi = NULL, mu, phi, type = "pdf", power = NULL, 
                    add = FALSE, 
                    plot_args = list(), point_args = list(), line_args = list())

tweedie.plot(
  y,
  xi = NULL,
  mu,
  phi,
  type = "pdf",
  power = NULL,
  add = FALSE,
  ...
)
tweedie_plot(y, xi = NULL, mu, phi, type = "pdf", power = NULL, 
                    add = FALSE, 
                    plot_args = list(), point_args = list(), line_args = list())

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

Arguments

y

the values for $y$ in the plot.

xi

a synonym for power.

mu

the mean of the distribution $\mu$ .

phi

the dispersion parameter $\phi$ .

type

the type of plot, either pdf (the default) or cdf.

power

the variance power $p$ .

add

logical; if TRUE, the plot is added to the current plot; if FALSE (the default) the plot is produced on a fresh plot.

plot_args

A named list of arguments controlling the main plot. These are passed to plot.

point_args

A named list of graphical parameters for plotted points. These are passed to points.

line_args

A named list of graphical parameters for plotted lines. These are passed to lines.

...

Additional graphical arguments, passed to plot_args for backward compatibility. Deprecated; use plot_args in tweedie_plot instead.

Details

If $1 < p < 2$ , the mass at $Y=0$ is automatically added.

Examples

y <- seq(0, 4, length = 50)
tweedie_plot(y, power = 1.1, mu = 1, phi = 1,
             line_args = list(lwd = 2))


y <- seq(0, 4, length = 50)
tweedie_plot(y, power = 1.1, mu = 1, phi = 1,
             line_args = list(lwd = 2))

Profile Likelihood Estimate of Tweedie Variance Index Parameter

Description

This function profiles the (log-)likelihood over a vector of Tweedie power-index parameter (denoted $p$ or $\xi$ ) to find the maximum likelihood estimate (MLE) of the index parameter $p$ (or equivalently $\xi$ ).

Usage

tweedie_profile(formula, p.vec = NULL, xi.vec = NULL, link.power = 0, 
  data, weights = 1, offset = 0, fit.glm = FALSE, do.smooth = TRUE, 
  do.plot = FALSE, do.ci = do.smooth, eps = 1/6, 
  control = list( epsilon = 1e-09, maxit = stats::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,
  plot_args = list(), point_args = list(), line_args = list())

tweedie.profile(
  formula,
  p.vec = NULL,
  xi.vec = NULL,
  link.power = 0,
  data,
  weights = 1,
  offset = 0,
  fit.glm = FALSE,
  do.smooth = TRUE,
  do.plot = FALSE,
  do.ci = do.smooth,
  eps = 1/6,
  control = list(epsilon = 1e-09, maxit = stats::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
)
tweedie_profile(formula, p.vec = NULL, xi.vec = NULL, link.power = 0, 
  data, weights = 1, offset = 0, fit.glm = FALSE, do.smooth = TRUE, 
  do.plot = FALSE, do.ci = do.smooth, eps = 1/6, 
  control = list( epsilon = 1e-09, maxit = stats::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,
  plot_args = list(), point_args = list(), line_args = list())

tweedie.profile(
  formula,
  p.vec = NULL,
  xi.vec = NULL,
  link.power = 0,
  data,
  weights = 1,
  offset = 0,
  fit.glm = FALSE,
  do.smooth = TRUE,
  do.plot = FALSE,
  do.ci = do.smooth,
  eps = 1/6,
  control = list(epsilon = 1e-09, maxit = stats::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.

p.vec

a vector of $p$ values for consideration. The values must all be larger than one. If the response has zeros, values must be $1 < p < 2$ . If NULL (default), p.vec is set automatically.

xi.vec

a synonym for p.vec, as some authors use the $\xi$ notation.

link.power

the power link function to use in the tweedie glm family. These link functions $g(\cdot)$ are of the form $g(\eta)=\eta^{link.power}$ , where link.power = 0 (default) refers to the logarithm link function.

data

an optional data frame, list or environment containing the variables.

weights

an optional vector of weights to be used in the fitting process.

offset

an a priori known component included in the linear predictor. See model.offset.

fit.glm

logical; if TRUE, the Tweedie glm is fitted using the value of $p$ found by the profiling function. The default is FALSE.

do.smooth

logical; if TRUE (default), a spline is fitted to the data to smooth the profile likelihood plot. Note that p.vec must contain at least five points for smoothing.

do.plot

logical; if TRUE, a plot of the profile likelihood is produced. The default is FALSE.

do.ci

logical; if TRUE, the nominal 100*conf.level is computed. Defaults to the value of do.smooth. Confidence intervals are only computed if do.smooth = TRUE.

eps

the offset in computing the variance function. Default is 1/6 (as recommended by Nelder and Pregibon, 1987). eps is ignored unless method = "saddlepoint".

control

a list of parameters for controlling the fitting process;

do.points

logical; if TRUE, the points used to compute the likelihood as given by p.vec (or equivalently, xi.vec) are explicitly shown by points. The defaults is the value of do.plot.

method

the method of evaluation; one of saddlepoint, interpolation, series or inversion (the default).

conf.level

the level of confidence for the confidence intervals; the default is 0.95 (for $95\%$ confidence intervals).

phi.method

the method used to estimate $\phi$ ; one of saddlepoint, mle (the default).

verbose

logical; if TRUE, some details of the calculations are shown. The default is FALSE.

add0

logical; if TRUE, adds $P(Y = 0)$ to the plot. The default is FALSE.

plot_args

A named list of arguments controlling the main plot. These are passed to plot.

point_args

A named list of graphical parameters for plotted points. These are passed to points.

line_args

A named list of graphical parameters for plotted lines. These are passed to lines.

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 $p$ is found, the distribution within the class of Tweedie distributions is identified.

Note

The estimates of $p$ and $\phi$ are printed invisibly. If the response variable has any exact zeros, the values in p.vec must all be between one and two.

The function can be temperamental (for theoretical reasons involved in numerically computing the density; see Dunn and Smyth (2005)) and may be very slow or fail. One solution is to change the method. The default is method = "inversion"; then try "series", "interpolation", and "saddlepoint" in that order. Note that method = "saddlepoint" is an approximate method only.

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.

References

Dunn, P. K. and Smyth, G. K. (2018). Generalized linear models with examples in R. Springer. doi:10.1007/978-1-4419-0118-7

Examples

data(rock)
out <- tweedie_profile(perm~1, data=rock, do.plot=FALSE, 
                       xi.vec=seq(1.5, 2.75, length=11),
                       line_arg = list(lwd = 2))

# The estimate for the variance power index (p, or xi) is:
out$p.max

data(rock)
out <- tweedie_profile(perm~1, data=rock, do.plot=FALSE, 
                       xi.vec=seq(1.5, 2.75, length=11),
                       line_arg = list(lwd = 2))

# The estimate for the variance power index (p, or xi) is:
out$p.max

Package 'tweedie'

Help Index

Evaluation of Tweedie Exponential Family Models

Description

Author(s)

Fourier Inversion Evaluation for the Tweedie Probability Function

Description

Usage

Arguments

Value

Note

References

Examples

Tweedie densities evaluation using the saddlepoint approximation

Description

Usage

Arguments

Value

References

Examples

Series Evaluation for the Tweedie Probability Function

Description

Usage

Arguments

Value

References

Examples

Log-likelihood for Tweedie distributions

Description

Usage

Arguments

Details

Value

Note

References

See Also

Examples

Fourier Inversion Evaluation for the Tweedie Distribution Function

Description

Usage

Arguments

Value

Note

References

Examples

Series Evaluation for the Tweedie Distribution Function

Description

Usage

Arguments

Value

Note

References

Examples

Tweedie distributions

Description

Usage

Arguments

Details

Value

Note

References

See Also

Examples

AIC for Tweedie Glms

Description

Usage

Arguments

Details

Value

Note

References

See Also

Examples

Tweedie Distribution: Convert Between Parameter Formats

Description

Usage

Arguments

Value

Examples

Unit Deviance for a Tweedie Distribution