conquer.reg {conquer} | R Documentation |

Fit sparse quantile regression models in high dimensions via regularized conquer methods with "lasso", "elastic-net", "group lasso", "sparse group lasso", "scad" and "mcp" penalties.
For "scad" and "mcp", the iteratively reweighted `\ell_1`

-penalized algorithm is complemented with a local adpative majorize-minimize algorithm.

```
conquer.reg(
X,
Y,
lambda = 0.2,
tau = 0.5,
kernel = c("Gaussian", "logistic", "uniform", "parabolic", "triangular"),
h = 0,
penalty = c("lasso", "elastic", "group", "sparse-group", "scad", "mcp"),
para.elastic = 0.5,
group = NULL,
para.scad = 3.7,
para.mcp = 3,
epsilon = 0.001,
iteMax = 500,
phi0 = 0.01,
gamma = 1.2,
iteTight = 3
)
```

`X` |
An |

`Y` |
An |

`lambda` |
( |

`tau` |
( |

`kernel` |
( |

`h` |
( |

`penalty` |
( |

`para.elastic` |
( |

`group` |
( |

`para.scad` |
( |

`para.mcp` |
( |

`epsilon` |
( |

`iteMax` |
( |

`phi0` |
( |

`gamma` |
( |

`iteTight` |
( |

An object containing the following items will be returned:

`coeff`

If the input

`lambda`

is a scalar, then`coeff`

returns a`(p + 1)`

vector of estimated coefficients, including the intercept. If the input`lambda`

is a sequence, then`coeff`

returns a`(p + 1)`

by`nlambda`

matrix, where`nlambda`

refers to the length of`lambda`

sequence.`bandwidth`

Bandwidth value.

`tau`

Quantile level.

`kernel`

Kernel function.

`penalty`

Penalty type.

`lambda`

Regularization parameter(s).

`n`

Sample size.

`p`

Number of the covariates.

Belloni, A. and Chernozhukov, V. (2011). `\ell_1`

penalized quantile regression in high-dimensional sparse models. Ann. Statist., 39, 82-130.

Fan, J. and Li, R. (2001). Variable selection via nonconcave regularized likelihood and its oracle properties. J. Amer. Statist. Assoc., 96, 1348-1360.

Fan, J., Liu, H., Sun, Q. and Zhang, T. (2018). I-LAMM for sparse learning: Simultaneous control of algorithmic complexity and statistical error. Ann. Statist., 46, 814-841.

Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33-50.

Simon, N., Friedman, J., Hastie, T. and Tibshirani, R. (2013). A sparse-group lasso. J. Comp. Graph. Statist., 22, 231-245.

Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. R. Statist. Soc. Ser. B, 58, 267–288.

Tan, K. M., Wang, L. and Zhou, W.-X. (2022). High-dimensional quantile regression: convolution smoothing and concave regularization. J. Roy. Statist. Soc. Ser. B, 84, 205-233.

Yuan, M. and Lin, Y. (2006). Model selection and estimation in regression with grouped variables., J. Roy. Statist. Soc. Ser. B, 68, 49-67.

Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. Ann. Statist., 38, 894-942.

Zou, H. and Hastie, T. (2005). Regularization and variable selection via the elastic net. J. R. Statist. Soc. Ser. B, 67, 301-320.

See `conquer.cv.reg`

for regularized quantile regression with cross-validation.

```
n = 200; p = 500; s = 10
beta = c(rep(1.5, s), rep(0, p - s))
X = matrix(rnorm(n * p), n, p)
Y = X %*% beta + rt(n, 2)
## Regularized conquer with lasso penalty at tau = 0.7
fit.lasso = conquer.reg(X, Y, lambda = 0.05, tau = 0.7, penalty = "lasso")
beta.lasso = fit.lasso$coeff
## Regularized conquer with elastic-net penalty at tau = 0.7
fit.elastic = conquer.reg(X, Y, lambda = 0.1, tau = 0.7, penalty = "elastic", para.elastic = 0.7)
beta.elastic = fit.elastic$coeff
## Regularized conquer with scad penalty at tau = 0.7
fit.scad = conquer.reg(X, Y, lambda = 0.13, tau = 0.7, penalty = "scad")
beta.scad = fit.scad$coeff
## Regularized conquer with group lasso at tau = 0.7
beta = c(rep(1.3, 5), rep(1.5, 5), rep(0, p - s))
err = rt(n, 2)
Y = X %*% beta + err
group = c(rep(1, 5), rep(2, 5), rep(3, p - s))
fit.group = conquer.reg(X, Y, lambda = 0.05, tau = 0.7, penalty = "group", group = group)
beta.group = fit.group$coeff
```

[Package *conquer* version 1.3.0 Index]