# QBIC: High dimensional BIC for quantile regression model In QICD: Estimate the Coefficients for Non-Convex Penalized Quantile Regression Model by using QICD Algorithm

## Description

A high dimensional BIC will be returned specificall for quantile regression

## Usage

 `1` ```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

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 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

`checkloss`, `QICD`
 ``` 1 2 3 4 5 6 7 8 9 10``` ```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) ```