wle (version 0.5)

wle.gamma: Robust Estimation in the Gamma model

Description

wle.gamma is a preliminary version; it 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.

Usage

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), shape.tol=10, 
          use.smooth=TRUE, tol.int)

Arguments

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.

shape.int

a 2 dimension vector for the interval search of the shape parameter at the first iteration.

shape.tol

the length of the interval search used in the successive iterations for 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 $tol*10^{-4}$.

Value

• 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.
• 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.
• 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.

Details

The gamma is parametrized as follows ($\alpha = scale$, $\omega = shape$):

$f(x) = 1/(\alpha^\omega Gamma(\omega)) x^(\omega-1) e^-(x/\alpha)$

for $x > 0$, $\alpha > 0$ and $\omega > 0$.

The function use uniroot to solve the estimating equation for $shape$, hence you can expect error from it. You can use shape.int and shape.tol to avoid them. It also use a fortran routine (dqagp) to calculate the smoothed model, i.e., evaluate an 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 $smooth*shape/scale^2$.

References

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.

Examples

library(wle)

set.seed(1234)

x_rgamma(100,2,2)

wle.gamma(x)

x_c(rgamma(30,2,2),rgamma(100,20,20))

wle.gamma(x, boot=10, group=10, num.sol=2)