High dimensional BIC for quantile regression model

Description

A high dimensional BIC will be returned specificall for quantile regression

Usage

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QBIC(y, X, beta, tau = 0.5, const = 6)

Arguments

y

response y as in QICD.

X

x matrix as in QICD.

beta

the coefficients vector for BIC calculation

tau

tau value as in QICD

const

a constant to adjust the BIC. A positive numerical value; default value is 6.

Details

The high dimensional BIC for quantile regression model is

log(checkloss)+|S|log(log(n))C_n/n

where S is the selected model in QICD, n is the number of obs, C_n is some positive constant which diverges to infinity as n increases. Actually, C_n is log(p)/const.

Value

QBIC will be returned, which is a numerical value

Author(s)

Bo Peng

References

Lee, E. R., Noh, H. and Park. B. (2013) Model Selection via Bayesian Information Criterion for Quantile Regression Models. Journal of the American Statistical Associa- tion, preprint. http://www.tandfonline.com/doi/pdf/10.1080/01621459.2013.836975 \Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1080/01621459.2013.836975")}

Wang,L., Kim, Y., and Li,R. (2013+) Calibrating non-convex penalized regression in ultra-high dimension. To appear in Annals of Statistics. http://users.stat.umn.edu/~wangx346/research/nonconvex.pdf

See Also

checkloss, QICD

Examples

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x=matrix(rnorm(1000),50)
n=dim(x)[1]
p=dim(x)[2]
intercept=1
y=x[,1]+x[,7]+x[,9]+0.1*rnorm(n)
beta1=rep(0,p+intercept)
tau=0.5
a=2.7
res=QICD(y,x,beta1,tau,lambda=10,a,"scad",intercept=intercept)
QBIC(y,cbind(x,rep(1,n)),res$beta_final,tau=tau)