cqr.lasso.mm {cqrReg}

Composite Quantile Regression (cqr) with Adaptive Lasso Penalty (lasso) use Majorize and Minimize (mm) Algorithm

Description

The adaptive lasso penalty parameter base on the estimated coefficient without penalty function. Composite quantile regression find the estimated coefficient which minimize the absolute error for various quantile level. The algorithm majorizing the objective function by a quadratic function followed by minimizing that quadratic.

Usage

cqr.lasso.mm(X,y,tau,lambda,beta,maxit,toler)

Arguments

X

the design matrix

y

response variable

tau

vector of quantile level

lambda

The constant coefficient of penalty function. (default lambda=1)

beta

initial value of estimate coefficient (default naive guess by least square estimation)

maxit

maxim iteration (default 200)

toler

the tolerance critical for stop the algorithm (default 1e-3)

Value

a list structure is with components

beta

the vector of estimated coefficient

b

intercept for various quantile level

Note

cqr.lasso.mm(x,y,tau) work properly only if the least square estimation is good.

References

David R.Hunter and Runze Li.(2005) Variable Selection Using MM Algorithms,The Annals of Statistics 33, Number 4, Page 1617–1642.

Hui Zou and Ming Yuan(2008). Composite Quantile Regression and the Oracle Model Selection Theory, The Annals of Statistics, 36, Number 3, Page 1108–1126.

Examples

set.seed(1)
n=100
p=2
a=2*rnorm(n*2*p, mean = 1, sd =1)
x=matrix(a,n,2*p)
beta=2*rnorm(p,1,1)
beta=rbind(matrix(beta,p,1),matrix(0,p,1))
y=x%*%beta-matrix(rnorm(n,0.1,1),n,1)
tau=1:5/6
# x is 1000*20 matrix, y is 1000*1 vector, beta is 20*1 vector with last ten zero value elements. 
cqr.lasso.mm(x,y,tau)