Tweedie Distributions: mle estimation of p

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Description

Maximum likelihood estimation of the Tweedie index parameter power.

Usage

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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() are of the form g(eta) = eta^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, often 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 lohg-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 tempermental (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.

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.

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

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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)