GSboot_multireg function

Fast and Robust Bootstrap for GS-Estimates

Calculates bootstrapped GS-estimates and bootstrap confidence intervals using the Fast and Robust Bootstrap method.

GSboot_multireg(X, Y, R = 999, conf=0.95, ests = GSest_multireg(X, Y))

Arguments

• X: a matrix or data frame containing the explanatory variables (possibly including intercept).
• Y: a matrix or data frame containing the response variables.
• R: number of bootstrap samples. Default is R=999.
• conf: confidence level of the bootstrap confidence intervals. Default is conf=0.95.
• ests: GS-estimates as returned by GSest_multireg().

Details

Called by FRBmultiregGS and typically not to be used on its own. If no original GS-estimates are provided the function calls GSest_multireg with its default settings.

The fast and robust bootstrap was first introduced by Salibian-Barrera and Zamar (2002) for univariate regression MM-estimators and developed for GS-estimates by Roelant et al. (2009).

The value centered gives a matrix with R columns and $p*q+q*q$ rows ($p$ is the number of explanatory variables and $q$ is the number of response variables), containing the recalculated GS-estimates. Each column represents a different bootstrap sample. The first $p*q$ rows are the recalculated coefficient estimates and the next $q*q$ rows are the covariance estimates (the estimates are vectorized, i.e. columns stacked on top of each other). These bootstrap estimates are centered by the original estimates, which are also returned through vecest in vectorized form.

The output list further contains bootstrap standard errors, as well as so-called basic bootstrap confidence intervals and bias corrected and accelerated confidence intervals (Davison and Hinkley, 1997, p.194 and p.204 respectively). Also in the output are p-values defined as 1 minus the smallest confidence level for which the confidence intervals would include the (hypothesised) value of zero. Both BCa and basic bootstrap p-values are given. These are only useful for the regression coefficient estimates (not really for the covariance estimates).

Bootstrap samples which contain too few distinct observations with positive weights are discarded (a warning is given if this happens). The number of samples actually used is returned via ROK.

Returns

A list containing the following components: - centered: a matrix of all fast and robust bootstrap recalculations where the recalculations are centered by the original estimates (see Details)

• vecest: a vector containing the orginal estimates stacked on top of each other

• SE: bootstrap standard errors for the estimates in vecest

• cov: bootstrap covariance matrix for the estimates in vecest

• CI.bca: a matrix containing bias corrected and accelerated confidence intervals, corresponding to the estimates in vecest (first column are lower limits, second column are upper limits)

• CI.basic: a matrix containing basic bootstrap intervals, corresponding to the estimates in vecest (first column are lower limits, second column are upper limits)

• p.bca: a vector containing p-values based on the bias corrected and accelerated confidence intervals (corresponding to the estimates in vecest)

• p.basic: a vector containing p-values based on the basic bootstrap intervals (corresponding to the estimates in vecest)

• ROK: number of bootstrap samples actually used (i.e. not discarded due to too few distinct observations with positive weight)

References

• A.C. Davison, D.V. Hinkley (1997) Bootstrap methods and their application. Cambridge University Press.
• E. Roelant, S. Van Aelst and C. Croux (2009) Multivariate Generalized S-estimators. Journal of Multivariate Analysis, 100 , 876--887.
• M. Salibian-Barrera, S. Van Aelst and G. Willems (2008) Fast and robust bootstrap. Statistical Methods and Applications, 17 , 41--71.
• M. Salibian-Barrera, R.H. Zamar (2002) Bootstrapping robust estimates of regression. The Annals of Statistics, 30 , 556--582.
• 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, Stefan Van Aelst and Gert Willems

See Also

FRBmultiregGS, GSest_multireg

Examples

data(schooldata)
school.x1 <- data.matrix(schooldata[,1:2])
school.y <- data.matrix(schooldata[,6:8])

## computes 10 bootstrap recalculations starting from the GS-estimator
## obtained from GSest_multireg

    bootres <- GSboot_multireg(school.x1,school.y,R=5)