Tweedie Distributions
Derivatives of the log-likelihood with respect to phi
dtweedie.dldphi(phi, mu, power, y ) dtweedie.dldphi.saddle(phi, mu, power, y )
y |
vector of quantiles |
mu |
the mean |
phi |
the dispersion |
power |
the value of power such that the variance is var(Y) = phi * mu^power |
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
var(Y) = phi * mu^power
where power is greater than or equal to one, or less than or equal to zero.
This function only evaluates for power
greater than or equal to one.
Special cases include the
normal (power=0),
Poisson (power=1 with phi=1),
gamma (power=2)
and
inverse Gaussian (power=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 value of the derivative
d(l)/d(phi)
where l 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.
Peter Dunn (pdunn2@usc.edu.au)
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.
### Plot dl/dphi against candidate values of phi
power <- 2
mu <- 1
phi <- seq(2, 8, by = 0.1)
set.seed(10000) # For reproducibility
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") )
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