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 pdunn2@usc.edu.au

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)

Arguments

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:

1. Evaluation of an infinite series (dtweedie_series).

2. 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.

See Also

dtweedie_series, dtweedie_inversion, ptweedie_series, ptweedie_inversion, dtweedie_saddle, tweedie_lambda

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)

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)

Arguments

Value

A numeric vector of densities if details=FALSE; if details=TRUE, return a list with density (the density values), regions (the number of integration regions used) and methods (which of the three methods was used).

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

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")

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)

Arguments

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")

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)

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

Arguments

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")

Log-likelihood for Tweedie distributions

Description

Evaluates the log-likelihood for a fitted Tweedie glm.

Usage

logLiktweedie(glm.obj, dispersion = NULL)

Arguments

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.

See Also

dtweedie

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)

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)

Arguments

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) and regions (a vector of the number of integration regions used).

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")

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)

Arguments

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")

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)

Arguments

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.

See Also

dtweedie

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)

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)

Arguments

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)

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)

Arguments

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)

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, yLimits = NULL)

Arguments

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 \sin\Im k(t). 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 (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.

See Also

dtweedie

Examples

tweedie_integrand(2, power = 3, mu = 1, phi = 1)

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)

Arguments

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)

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, ...)

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

Arguments

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)

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)

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

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(Loblolly)
out <- tweedie_profile(height~age, data = Loblolly, 
          do.plot = FALSE, p.vec = seq(3.5, 4.5, length = 7) )
# The estimate for the variance power index (p, or xi) is:
out$p.max