wle (version 0.9-91)

wle.normal.mixture: Robust Estimation in the Normal Mixture Model

Description

wle.normal.mixture is a preliminary version; it is used to robust estimate the location, scale and proportion parameters via Weighted Likelihood, when the sample is iid from a normal mixture univariate distribution with known m number of components.

Usage

wle.normal.mixture(x, m, boot=5, group, num.sol=1, raf="HD",  smooth=0.003, tol=10^(-15), equal=10^(-2),  max.iter=1000, all.comp=TRUE, min.size=0.02, min.weights=0.3, boot.start=10, group.start=3,  tol.start=10^(-6), equal.start=10^(-3), smooth.start=0.003, max.iter.start=500,  max.iter.boot=25, verbose=FALSE)
wle.normal.mixture.start(x, m, boot=5, group, raf="HD",  smooth=0.003, tol=10^(-15), equal=10^(-2),  min.size=0.02, min.weights=0.3, boot.start=20,  group.start=3, max.iter.start=500,  max.iter.boot=20, verbose=FALSE)

Arguments

x

a vector contain the observations.

m

numbers of components.

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 $max(round(size/4),2)$ where $size$ is the number of observations.

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.

all.comp

try to find all the components.

min.size

see details

min.weights

see details

boot.start

the number of starting points for the starting process.

group.start

the dimension of the bootstap subsamples in the starting process. The default value is $max(round(group/4),2)$.

tol.start

the absolute accuracy to be used to achieve convergence of the algorithm in the starting process.

equal.start

the absolute value for which two roots are considered the same in the starting process. (This parameter must be greater than tol.start).

smooth.start

the value of the smoothing parameter in the starting process.

max.iter.start

maximum number of iterations in the starting process.

max.iter.boot

maximum number of iterations of the starting process.

verbose

if TRUE warnings are printed.

Value

wle.normal.mixture returns an object of class "wle.normal.mixture".Only print method is implemented for this class.The objects returned by wle.normal.mixture are:

location

the estimator of the location parameters, one vector for each root found.

scale

the estimator of the scale parameters, one vector for each root found.

pi

the estimator of the proportion parameters, one vector 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.

freq

the number of starting points converging to the roots.

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.

call

the match.call().

Details

this function use an iterative procedure to evaluate starting points. First, using wle.normal we try to find the biggest components, then we discard each observation with weight greater than min.weights. The wle.normal is run on the remain observations if the ratio between the number of observations and the original sample size is greater than min.size. The convergence of the algorithm is determined by the difference between two iterations. This stopping rule could have some problems, as soon as possible it will replace with the one proposed in Markatou (2000) pag. 485 (5).

References

Markatou, M., (2000) Mixture models, robustness and the weighted likelihood methodology, Biometrics, 56, 483-486.

Markatou, M., (2001) A closer look at the weighted likelihood in the context of mixtures, Probability and Statistical Models with Applications, Charalambides, C.A., Koutras, M.V. and Balakrishnan, N. (eds.), Chapman and Hall/CRC, 447-467.

Examples

library(wle)
set.seed(1234)
x <- c(rnorm(150,0,1),rnorm(50,15,2))
wle.normal.mixture(x,m=2,group=50,group.start=2,boot=5,num.sol=3)
wle.normal(x,group=2,boot=10,num.sol=3)