Mqreg | R Documentation |

Robust M-quantiles are estimated using an iterative penalised reweighted least squares approach. Effects using quadratic penalties can be included, such as P-splines, Markov random fields or Kriging.

Mqreg(formula, data = NULL, smooth = c("schall", "acv", "fixed"), estimate = c("iprls", "restricted"),lambda = 1, tau = NA, robust = 1.345, adaptive = FALSE, ci = FALSE, LSMaxCores = 1)

`formula` |
An R formula object consisting of the response variable, '~'
and the sum of all effects that should be taken into consideration.
Each effect has to be given through the function |

`data` |
Optional data frame containing the variables used in the model, if the data is not explicitely given in the formula. |

`estimate` |
Character string defining the estimation method that is used to fit the expectiles. Further detail on all available methods is given below. |

`smooth` |
There are different smoothing algorithms that should prevent overfitting.
The 'schall' algorithm iterates the smoothing penalty |

`lambda` |
The fixed penalty can be adjusted. Also serves as starting value for the smoothing algorithms. |

`tau` |
In default setting, the expectiles (0.01,0.02,0.05,0.1,0.2,0.5,0.8,0.9,0.95,0.98,0.99) are calculated.
You may specify your own set of expectiles in a vector. The option may be set to 'density' for the calculation
of a dense set of expectiles that enhances the use of |

`robust` |
Robustness constant in M-estimation. See |

`adaptive` |
Logical. Whether the robustness constant is adapted along the covariates. |

`ci` |
Whether a covariance matrix for confidence intervals and the summary function is calculated. |

`LSMaxCores` |
How many cores should maximal be used by parallelization |

In the least squares approach the following loss function is minimised:

* S = ∑_{i=1}^{n}{ w_p(y_i - m_i(p))^2} *

with weights

* w_p(u) = (-(1-p)*c*(u_i< -c)+(1-p)*u_i*(u_i<0 \& u_i>=-c)+p*u_i*(u_i>=0 \& u_i<c)+p*c*(u_i>=c)) / u_i *

for quantiles and

* w_p(u) = -(1-p)*c*(u_i< -c)+(1-p)*u_i*(u_i<0 \& u_i>=-c)+p*u_i*(u_i>=0 \& u_i<c)+p*c*(u_i>=c) *

for expectiles, with standardised residuals *u_i = 0.6745*(y_i - m_i(p)) / median(y-m(p))* and robustness constant c.

An object of class 'expectreg', which is basically a list consisting of:

`lambda ` |
The final smoothing parameters for all expectiles and for all effects in a list. For the restricted and the bundle regression there are only the mean and the residual lambda. |

`intercepts ` |
The intercept for each expectile. |

`coefficients` |
A matrix of all the coefficients, for each base element a row and for each expectile a column. |

`values` |
The fitted values for each observation and all expectiles, separately in a list for each effect in the model, sorted in order of ascending covariate values. |

`response` |
Vector of the response variable. |

`covariates` |
List with the values of the covariates. |

`formula` |
The formula object that was given to the function. |

`asymmetries` |
Vector of fitted expectile asymmetries as given by argument |

`effects` |
List of characters giving the types of covariates. |

`helper` |
List of additional parameters like neighbourhood structure for spatial effects or 'phi' for kriging. |

`design` |
Complete design matrix. |

`fitted` |
Fitted values |

`plot`

, `predict`

, `resid`

,
`fitted`

, `effects`

and further convenient methods are available for class 'expectreg'.

Monica Pratesi

University Pisa

https://www.unipi.it

M. Giovanna Ranalli

University Perugia

https://www.unipg.it

Nicola Salvati

University Perugia

https://www.unipg.it

Fabian Otto-Sobotka

University Oldenburg

https://uol.de

Pratesi M, Ranalli G and Salvati N (2009)
*Nonparametric M-quantile regression using penalised splines*
Journal of Nonparametric Statistics, 21:3, 287-304.

Otto-Sobotka F, Ranalli G, Salvati N, Kneib T (2019)
*Adaptive Semiparametric M-quantile Regression*
Econometrics and Statistics 11, 116-129.

`expectreg.ls`

, `rqss`

data("lidar", package = "SemiPar") m <- Mqreg(logratio~rb(range,"pspline"),data=lidar,smooth="f", tau=c(0.05,0.5,0.95),lambda=10) plot(m,rug=FALSE)

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