wle.gamma: Robust Estimation in the Gamma model
In wle: Weighted Likelihood Estimation

Description Usage Arguments Details Value Author(s) References Examples

wle.gamma is used to robust estimate the shape and the scale parameters via Weighted Likelihood, when the majority of the data are from a gamma distribution.

wle.gamma(x, boot=30, group, num.sol=1, raf="HD", smooth=0.008,
          tol=10^(-6), equal=10^(-3), max.iter=500,
          shape.int=c(0.01, 100), use.smooth=TRUE, tol.int,
          verbose=FALSE, maxiter=1000)

`x`	a vector contain the observations.
`boot`	the number of starting points based on boostrap subsamples to use in the search of the roots.
`group`	the dimension of the bootstap subsamples. The default value is max(round(size/4),var) where size is the number of observations and var is the number of variables.
`num.sol`	maximum number of roots to be searched.
`raf`	type of Residual adjustment function to be use: `raf="HD"`: Hellinger Distance RAF, `raf="NED"`: Negative Exponential Disparity RAF, `raf="SCHI2"`: Symmetric Chi-Squared Disparity RAF.
`smooth`	the value of the smoothing parameter.
`tol`	the absolute accuracy to be used to achieve convergence of the algorithm.
`equal`	the absolute value for which two roots are considered the same. (This parameter must be greater than `tol`).
`max.iter`	maximum number of iterations for the main function.
`shape.int`	a 2 dimension vector for the interval search of the shape parameter.
`use.smooth`	if `FALSE` the unsmoothed model is used. This is usefull when the integration routine does not work well.
`tol.int`	the absolute accuracy to be used in the integration routine. The default value is tol10^{-4}*.
`verbose`	if `TRUE` warnings are printed.
`maxiter`	maximum number of iterations. This value is passed to `uniroot` function.

The gamma is parametrized as follows (α = scale, ω = shape):

f(x) = 1/(α^ω Gamma(ω)) x^(ω-1) e^-(x/α)

for x > 0, α > 0 and ω > 0.

The function use uniroot to solve the estimating equation for shape, errors from uniroot are handled by try. If errors occurs then the function returns NA.

You can use shape.int to avoid them. It also use a fortran routine (dqagp) to calculate the smoothed model, i.e., evaluate the integral. Sometime the accuracy is not satisfactory, you can use use.smooth=FALSE to have an approximate estimation using the model instead of the smoothed model.

The Folded Normal distribution is use as kernel. The bandwith is smooth*shape/scale^2.

wle.gamma returns an object of class "wle.gamma".

Only print method is implemented for this class.

The object returned by wle.gamma are:

`shape`	the estimator of the shape parameter, one value for each root found.
`scale`	the estimator of the scale parameter, one value for each root found.
`rate`	the estimator of the rate parameter (1/scale), one value for each root found.
`tot.weights`	the sum of the weights divide by the number of observations, one value for each root found.
`weights`	the weights associated to each observation, one column vector for each root found.
`f.density`	the non-parametric density estimation.
`m.density`	the smoothed model.
`delta`	the Pearson residuals.
`call`	the match.call().
`tot.sol`	the number of solutions found.
`not.conv`	the number of starting points that does not converge after the `max.iter` iteration are reached.

Claudio Agostinelli

Markatou, M., Basu, A. and Lindsay, B.G., (1998). Weighted likelihood estimating equations with a bootstrap root search, Journal of the American Statistical Association, 93, 740-750.

Agostinelli, C., (1998). Inferenza statistica robusta basata sulla funzione di verosimiglianza pesata: alcuni sviluppi, Ph.D Thesis, Department of Statistics, University of Padova.

library(wle)

x <- rgamma(n=100, shape=2, scale=2)

wle.gamma(x)

x <- c(rgamma(n=30, shape=2, scale=2), rgamma(n=100, shape=20, scale=20))

wle.gamma(x, boot=10, group=10, num.sol=2) # depending on the sample, one or two roots.