Package {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 pdunn2@usc.edu.au

Authors:

• 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, log = FALSE)
ptweedie(q, xi = NULL, mu, phi, power = NULL, lower.tail = TRUE, 
  log.p = FALSE, verbose = FALSE)
qtweedie(p, xi = NULL, mu, phi, power = NULL)
rtweedie(n, xi = NULL, mu, phi, power = NULL)

Arguments

Details

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. For the density, dtweedie() uses one of three methods for each value, depending on p and z = \phi y^{p - 2} (the dispersion after rescaling y to one):

1. the saddlepoint approximation with its asymptotic correction, where z is small enough for its estimated relative error to be below 10^{-13};

2. otherwise, the series (dtweedie_series) for 1 < p < 2, and for p > 2 where its estimated relative error is below 10^{-12};

3. otherwise, the Fourier inversion (dtweedie_inversion).

Against high-precision reference values, the relative error of the density is below about 4 \times 10^{-13}. (Before version 3.2.0, dtweedie() also interpolated from stored values of the Fourier inversion, which was less accurate.)

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

Value

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

As for R's own functions for distributions (such as dnorm), the result of dtweedie(), ptweedie() and qtweedie() has the same length and form as the first input (y, q or p): its attributes are kept, so (for example) a matrix gives a matrix, and a named vector gives a named vector. (mu and phi must be scalars, or the same length as the first input.) rtweedie() returns a vector of length n, without attributes.

Note

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

References

Dunn, Peter K and Smyth, Gordon K (Submitted). Evaluation of the Tweedie exponential dispersion model distribution function. Submitted.

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, a list containing density (a vector of the values of the density), logdensity (its logarithm, computed without underflow), 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 statmod::[pdpqr]invgauss, all of which are very accurate (Giner & Smyth, 2016). 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

Giner G., Smyth G. K. (2016). statmod: probability calculations for the inverse Gaussian distribution. The R Journal, 8(1), 339–351. doi:10.32614/RJ-2016-024

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.

Accuracy

The saddlepoint approximation is an approximation, not an exact evaluation, and no warning is given when it is inaccurate. Its relative error grows with z = \phi y^{p - 2}, and is roughly of order z/10 or smaller (for example, exactly z/12 to first order for the gamma distribution, p = 2); it is small only when z is small. For 1 < p < 2, the default offset eps = 1/6 (which keeps the density finite near y = 0) adds a further relative error of about (1 + \epsilon/y)^{-p/2} - 1, which does not shrink as \phi becomes small (about 11% at y = 1 when p = 1.5).

dtweedie() uses a corrected saddlepoint approximation only where its estimated relative error is below 10^{-13}; use dtweedie() for accurate 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, details = FALSE)

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

Arguments

Value

A numeric vector of probabilities.

Accuracy

For 1 < p < 2, the terms of the series are all positive, and the density keeps full relative accuracy. For p > 2, the terms alternate in sign, and rounding error can cancel many digits. An estimate of the resulting relative error is returned (as relerr) when details = TRUE; no warning is given for a large estimated error. Where the series cannot be summed at all, NaN is returned with a warning. dtweedie() uses the series for p > 2 only where this estimated relative error is below 10^{-12}.

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, lower.tail = TRUE, log.p = FALSE, IGexact = TRUE, 
                          verbose = FALSE, details = FALSE)

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), regions (a vector of the number of integration regions used), exitstatus (a vector, where a 1 for any value means a computational problem or target relative accuracy not reached, for the corresponding observation), and small_tail (a logical vector; TRUE where the inversion returned a tail probability below 10^{-8}, whose relative accuracy may be poor; see the Accuracy section).

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.

Accuracy

The inversion computes F(y) = 1/2 - I/\pi (and 1 - F(y) = 1/2 + I/\pi for the upper tail), where I is the integral evaluated numerically (for 1 < p < 2, the constant 1/2 is replaced by (1 \pm e^{-\lambda})/2). When the requested tail probability P is small, I/\pi is close to 1/2 and most of the digits cancel, so the error is absolute (a few times 10^{-16}) rather than relative: the relative error is about 10^{-16}/P or a few times larger. Tail probabilities below about 10^{-8} may therefore have poor relative accuracy, and those far below 10^{-16} may have no correct digits. These values are flagged by small_tail when details = TRUE; no warning is given.

ptweedie() does not have this limitation: it recomputes small tail probabilities from the inversion by integrating the density, which keeps its relative accuracy far into the tails. Use ptweedie() for small tail probabilities.

Note

The 'exact' values for the inverse Gaussian distribution are not really exact, but evaluated using statmod::[pdpqr]invgauss, all of which are very accurate (Giner & Smyth, 2016).

References

Dunn, Peter K and Smyth, Gordon K (Submitted). Evaluation of the Tweedie exponential dispersion model distribution function. Submitted.

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

Giner G., Smyth G. K. (2016). statmod: probability calculations for the inverse Gaussian distribution. The R Journal, 8(1), 339–351. doi:10.32614/RJ-2016-024

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")

Tweedie CDF evaluation using the saddlepoint approximation

Description

Distribution function for the Tweedie EMDs using a saddlepoint approximation.

Usage

ptweedie_saddle(q, mu, phi, power, lower.tail = TRUE, log.p = FALSE)

Arguments

Value

A numeric vector of densities.

References

Dunn, Peter K and Smyth, Gordon K (Submitted). Evaluation of the Tweedie exponential dispersion model distribution function. Submitted.

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 distribution function
y <- seq(0.01, 4, length = 50)
Fy <- ptweedie_saddle(y, power = 2.1, mu = 1, phi = 1)
plot(y, Fy, type = "l", lwd = 2, ylab = "CDF")

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, lower.tail = TRUE, log.p = FALSE, 
                       verbose = FALSE, details = FALSE, max.terms = 1e6)

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

Arguments

Value

A numeric vector of probabilities (or log-probabilities, if log.p = TRUE).

Accuracy

The terms of the series are all positive, and are summed on the log scale, so both tails keep full relative accuracy, far into the tails. The only limitation is the number of terms: if more than max.terms are needed (when \lambda is very large), the series is truncated and a warning is given.

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 (Submitted). Evaluation of the Tweedie exponential dispersion model distribution function. Submitted.

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")

Tweedie CDF evaluation using direct integration (in the tails only)

Description

Distribution function evaluation in the distribution tails using direct integration of the density function.

Usage

ptweedie_tail_integrate(q, mu, phi, power, lower.tail = TRUE, IGexact = TRUE,
                               fallback = TRUE)

Arguments

References

Dunn, Peter K and Smyth, Gordon K (Submitted). Evaluation of the Tweedie exponential dispersion model distribution function. Submitted.

Examples

ptweedie_tail_integrate(0.00001, power = 2.1, mu = 1, phi = 1)

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

For 1 < p < 2, a Tweedie random variable is a Poisson sum of gamma random variables: Y = \sum_{i=1}^N Z_i, where N \sim \mbox{Pois}(\lambda) and the Z_i are independent gamma variables with shape \alpha and scale \gamma.

tweedie_convert() converts from the Tweedie (glm) parameters p, \mu and \phi to the underlying Poisson parameter \lambda and gamma parameters \alpha and \gamma.

tweedie_convert_back() converts the other way: from \lambda, \alpha and \gamma to p, \mu and \phi.

Usage

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

tweedie_convert_back(lambda, alpha, gamma)

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

Arguments

Details

The relationships are

\lambda = \frac{\mu^{2 - p}}{\phi (2 - p)}, \quad \alpha = \frac{2 - p}{p - 1}, \quad \gamma = \phi (p - 1) \mu^{p - 1}

and, in reverse,

p = \frac{\alpha + 2}{\alpha + 1}, \quad \mu = \lambda \alpha \gamma, \quad \phi = \frac{\lambda \alpha (\alpha + 1) \gamma^2}{\mu^p}.

Here \gamma is the gamma scale parameter; the corresponding rate parameter (sometimes written \beta) is \beta = 1/\gamma. In R's notation, the gamma variables are rgamma(n, shape = alpha, scale = gamma).

Value

tweedie_convert() returns a list of the parameters of the corresponding underlying Poisson and gamma distributions: poisson.lambda (\lambda from the underlying Poisson distribution), gamma.shape, gamma.scale (the shape \alpha and scale \gamma parameters from the underlying gamma distribution), p0 (the probability that Y = 0), gamma.mean and gamma.phi (the gamma mean \alpha\gamma and dispersion parameter 1/\alpha values).

tweedie_convert_back() returns a list containing mu (the Tweedie mean \mu), phi (the Tweedie dispersion parameter \phi), power (the Tweedie index parameter p), and p0 (the probability that Y = 0).

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)

### Convert, then convert back
pg <- tweedie_convert(mu = 2, phi = 0.5, power = 1.4)
tweedie_convert_back(lambda = pg$poisson.lambda,
                     alpha  = pg$gamma.shape,
                     gamma  = pg$gamma.scale)

Unit Deviance for a Tweedie Distribution

Description

Computes the unit deviance for Tweedie distributions. Since version 3.2.0 it is computed without cancellation, so it has full relative accuracy even when y is very close to mu.

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, 
                  plot_args  = list())

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 \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 (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,
                  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)

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, 
                    plot_args = list(), point_args = list(), line_args = list())

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,
             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,
  plot_args = list(),
  point_args = list(),
  line_args = list()
)

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 (or perhaps zero).

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)
# A quick profile, over a few values of p (no smoothing)
out <- tweedie_profile(perm ~ 1, data = rock, do.smooth = FALSE,
                       xi.vec = seq(1.8, 2.4, by = 0.2))
out$xi.max


# A finer profile, smoothed, with a confidence interval (slower)
out <- tweedie_profile(perm ~ 1, data = rock, do.plot = TRUE,
                       xi.vec = seq(1.5, 2.75, length = 11),
                       line_args = list(lwd = 2))

# The estimate for the variance power index (p, or xi), and its CI:
out$xi.max
out$ci