GSest_multireg function

GS Estimates for Multivariate Regression

Computes GS-Estimates of multivariate regression based on Tukey's biweight function.

## S3 method for class 'formula'
GSest_multireg(formula, data=NULL, ...)

## Default S3 method:
GSest_multireg(X, Y, int = TRUE, bdp = 0.5, control=GScontrol(...),
na.action=na.omit, ...)

Arguments

• formula: an object of class formula; a symbolic description of the model to be fit.
• data: data frame from which variables specified in formula are to be taken.
• X: a matrix or data frame containing the explanatory variables.
• Y: a matrix or data frame containing the response variables.
• int: logical: if TRUE an intercept term is added to the model (unless it is already present in X)
• bdp: required breakdown point. Should have $0 <$ bdp $\le 0.5$, the default is 0.5.
• control: a list with control parameters for tuning the computing algorithm, see GScontrol().
• na.action: a function which indicates what should happen when the data contain NAs. Defaults to na.omit.
• ...: allows for specifying control parameters directly instead of via control.

Details

Generalized S-estimators are defined by minimizing the determinant of a robust estimator of the scatter matrix of the differences of the residuals. Hence, this procedure is intercept free and only gives an estimate for the slope matrix. To estimate the intercept, we use the M-type estimator of location of Lopuhaa (1992) on the residuals with the residual scatter matrix estimate of the residuals as a preliminary estimate. We use a fast algorithm similar to the one proposed by Salibian-Barrera and Yohai (2006) for the regression case. See GScontrol for the adjustable tuning parameters of this algorithm.

The returned object inherits from class mlm such that the standard coef, residuals, fitted and predict functions can be used.

Returns

An object of class FRBmultireg which extends class mlm and contains at least the following components: - coefficients: GS-estimates of the regression coefficients

• residuals: the residuals, that is response minus fitted values

• fitted.values: the fitted values.

• Sigma: GS-estimate of the error covariance matrix

• Gamma: GS-estimate of the error shape matrix

• scale: GS-estimate of the size of the multivariate errors

• weights: implicit weights corresponding to the GS-estimates (i.e. final weights in the RWLS procedure for the intercept estimate)

• outFlag: outlier flags: 1 if the robust distance of the residual exceeds the .975 quantile of (the square root of) the chi-square distribution with degrees of freedom equal to the dimension of the responses; 0 otherwise

• b,c: tuning parameters used in Tukey biweight loss function, as determined by bdp

• method: a list with following components: est = character string indicating that GS-estimates were used, and bdp = a copy of the bdp argument

• control: a copy of the control argument

References

• H.P. Lopuhaa (1992) Highly efficient estimators of multivariate location with high breakdown point. The Annals of Statistics, 20 , 398-413.
• E. Roelant, S. Van Aelst and C. Croux (2009) Multivariate Generalized S-estimators. Journal of Multivariate Analysis, 100 , 876--887.
• M. Salibian-Barrera and V. Yohai (2006) A fast algorithm for S-regression estimates. Journal of Computational and Graphical Statistics, 15 , 414-427.
• S. Van Aelst and G. Willems (2013), Fast and robust bootstrap for multivariate inference: The R package FRB. Journal of Statistical Software, 53 (3), 1--32. tools:::Rd_expr_doi("10.18637/jss.v053.i03") .

Author(s)

Ella Roelant, Gert Willems and Stefan Van Aelst

See Also

FRBmultiregGS, GSboot_multireg, Sest_multireg, GScontrol

Examples

data(schooldata)
    school.x <- data.matrix(schooldata[,1:5])
    school.y <- data.matrix(schooldata[,6:8])
    GSest <- GSest_multireg(school.x,school.y,nsamp=50)
    
    ## or using the formula interface
    GSests <- GSest_multireg(cbind(reading,mathematics,selfesteem)~., data=schooldata)