Mqreg: Semiparametric M-Quantile Regression

In expectreg: Expectile and Quantile Regression

Semiparametric M-Quantile Regression

Description

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.

Usage

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)

Arguments

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 rb.
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 until it converges, the asymmetric cross-validation 'acv' minimizes a score-function using nlm or the function uses a fixed 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 cdf.qp and cdf.bundle afterwards.
robust	Robustness constant in M-estimation. See Details for definition.
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

Details

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.

Value

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 expectiles.
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 \hat{y} .

plot, predict, resid, fitted, effects and further convenient methods are available for class 'expectreg'.

Author(s)

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

References

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.

See Also

expectreg.ls, rqss

Examples

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)

expectreg documentation built on March 18, 2022, 5:57 p.m.