Weighted Mallows Cp
Description
The Weighted Mallows Cp is evaluated for each submodel.
Usage
1 2 3 4 5 
Arguments
formula 
a symbolic description of the model to be fit. The details of model specification are given below. 
data 
an optional data frame containing the variables
in the model. By default the variables are taken from
the environment which 
model, x, y 
logicals. If 
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. 
var.full 
the value of variance to be used in the denominator of the WCP, if 0 the variance estimated from the full model is used. 
num.sol 
maximum number of roots to be searched. 
raf 
type of Residual adjustment function to be use:

smooth 
the value of the smoothing parameter. 
tol 
the absolute accuracy to be used to achieve convergence of the algorithm. 
c
equal 
the absolute value for which two roots are considered the same. (This parameter must be greater than 
max.iter 
maximum number of iterations. 
min.weight 
see details. 
method 
see details. 
alpha 
penalty value. 
contrasts 
an optional list. See the 
verbose 
if TRUE warnings are printed. 
Details
Models for wle.cp
are specified symbolically. A typical model has the form response ~ terms
where response
is the (numeric) response vector and terms
is a series of terms which specifies a linear predictor for response
. A terms specification of the form first+second
indicates all the terms in first
together with all the terms in second
with duplicates removed. A specification of the form first:second
indicates the the set of terms obtained by taking the interactions of all terms in first
with all terms in second
. The specification first*second
indicates the cross of first
and second
. This is the same as first+second+first:second
.
min.weight
: the weighted likelihood equation could have more than one solution. These roots appear for particular situation depending on contamination level and type. The presence of multiple roots in the full model can create some problem in the set of weights we should use. Actually, the selection of the root is done by the minimum scale error provided. Since this choice is not always the one would choose, we introduce the min.weight
parameter in order to choose only between roots that do not down weight everything. This is not still the optimal solution, and perhaps, in the new release, this part will be change.
method
: this parameter, when set to "reduced"
, allows to use weights based on the reduced model. This is strongly discourage since the robust and asymptotic property of this kind of weighted Cp are not as good as the one based on method="full"
.
Value
wle.cp
returns an object of class
"wle.cp"
.
The function summary
is used to obtain and print a summary of the results.
The generic accessor functions coefficients
and residuals
extract coefficients and residuals returned by wle.cp
.
The object returned by wle.cp
are:
wcp 
Weighted Mallows Cp for each submodels 
coefficients 
the parameters estimator, one row vector for each root found and each submodel. 
scale 
an estimation of the error scale, one value for each root found and each submodel. 
residuals 
the unweighted residuals from the estimated model, one column vector for each root found and each submodel. 
tot.weights 
the sum of the weights divide by the number of observations, one value for each root found and each submodel. 
weights 
the weights associated to each observation, one column vector for each root found and each submodel. 
freq 
the number of starting points converging to the roots. 
call 
the match.call(). 
contrasts 

xlevels 

terms 
the model frame. 
model 
if 
x 
if 
y 
if 
info 
not well working yet, if 0 no error occurred. 
Author(s)
Claudio Agostinelli
References
Agostinelli, C., (1999). Robust model selection in regression via weighted likelihood methodology, Working Paper n. 1999.4, Department of Statistics, Universiy of Padova.
Agostinelli, C., (2002). Robust model selection in regression via weighted likelihood methodology, Statistics \& Probability Letters, 56, 289300.
Agostinelli, C., (1998). Inferenza statistica robusta basata sulla funzione di verosimiglianza pesata: alcuni sviluppi, Ph.D Thesis, Department of Statistics, University of Padova.
Agostinelli, C., Markatou, M., (1998). A onestep robust estimator for regression based on the weighted likelihood reweighting scheme, Statistics \& Probability Letters, Vol. 37, n. 4, 341350.
Agostinelli, C., (1998). Verosimiglianza pesata nel modello di regressione lineare, XXXIX Riunione scientifica della Societ\'a Italiana di Statistica, Sorrento 1998.
See Also
wle.smooth an algorithm to choose the smoothing parameter for normal distribution and normal kernel, wle.lm a function for estimating linear models with normal distribution error and normal kernel.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  library(wle)
x.data < c(runif(60,20,80),runif(5,73,78))
e.data < rnorm(65,0,0.6)
y.data < 8*log(x.data+1)+e.data
y.data[61:65] < y.data[61:65]4
z.data < c(rep(0,60),rep(1,5))
plot(x.data,y.data,xlab="X",ylab="Y")
xx.data < cbind(x.data,x.data^2,x.data^3,log(x.data+1))
colnames(xx.data) < c("X","X^2","X^3","log(X+1)")
result < wle.cp(y.data~xx.data,boot=10,group=10,num.sol=2)
summary(result)
plot(result,num.max=15)
result < wle.cp(y.data~xx.data+z.data,boot=10,group=10,num.sol=2)
summary(result)
plot(result,num.max=15)

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