boot.rq  R Documentation 
These functions can be used to construct standard errors, confidence intervals and tests of hypotheses regarding quantile regression models.
boot.rq(x, y, tau = 0.5, R = 200, bsmethod = "xy", mofn = length(y),
coef = NULL, blbn = NULL, cluster = NULL, U = NULL, ...)
x 
The regression design matrix 
y 
The regression response vector 
tau 
The quantile of interest 
R 
The number of bootstrap replications 
bsmethod 
The method to be employed. There are (as yet) five options: method = "xy" uses the xypair method, and method = "pwy" uses the method of Parzen, Wei and Ying (1994) method = "mcmb" uses the Markov chain marginal bootstrap of He and Hu (2002) and Kocherginsky, He and Mu (2003). The "mcmb" method isn't compatible with sparse X matrices. The fourth method = "wxy" uses the generalized bootstrap of Bose and Chatterjee (2003) with unit exponential weights, see also Chamberlain and Imbens (2003). The fifth method "wild" uses the wild bootstrap method proposed by Feng, He and Hu (2011). 
mofn 
optional argument for the bootstrap method "xy" that permits subsampling (m out of n) bootstrap. Obviously mofn should be substantially larger than the column dimension of x, and should be less than the sample size. 
coef 
coefficients from initial fitted object 
blbn 
orginal sample size for the BLB model 
cluster 
If nonNULL this argument should specify cluster id
numbers for each observation, in which case the clustered version of
the bootstrap based on the proposal of Hagemann (2017). If present

U 
If nonNULL this argument should specify an array of indices
or gradient evaluations to be used by the corresponding bootstrap
method as specified by 
... 
Optional arguments to control bootstrapping 
Their are several refinements that are still unimplemented. Percentile methods should be incorporated, and extensions of the methods to be used in anova.rq should be made. And more flexibility about what algorithm is used would also be good.
A list consisting of two elements:
A matrix B
of dimension R by p is returned with the R resampled
estimates of the vector of quantile regression parameters. When
mofn < n for the "xy" method this matrix has been deflated by
the factor sqrt(m/n).
A matrix U
of sampled indices (for bsmethod in c("xy", "wxy")
)
or gradient evaluations (for bsmethod in c("pwy", "cluster")
)
used to generate the bootstrapped realization, and potentially reused
for other taus
when invoked from summary.rqs
.
Roger Koenker (and Xuming He and M. Kocherginsky for the mcmb code)
[1] Koenker, R. W. (1994). Confidence Intervals for regression quantiles, in P. Mandl and M. Huskova (eds.), Asymptotic Statistics, 349–359, SpringerVerlag, New York.
[2] Kocherginsky, M., He, X. and Mu, Y. (2005). Practical Confidence Intervals for Regression Quantiles, Journal of Computational and Graphical Statistics, 14, 4155.
[3] Hagemann, A. (2017) Cluster Robust Bootstrap inference in quantile regression models, Journal of the American Statistical Association , 112, 446–456.
[4] He, X. and Hu, F. (2002). Markov Chain Marginal Bootstrap. Journal of the American Statistical Association , Vol. 97, no. 459, 783795.
[5] Parzen, M. I., L. Wei, and Z. Ying (1994): A resampling method based on pivotal estimating functions,” Biometrika, 81, 341–350.
[6] Bose, A. and S. Chatterjee, (2003) Generalized bootstrap for estimators of minimizers of convex functions, J. Stat. Planning and Inf, 117, 225239.
[7] Chamberlain G. and Imbens G.W. (2003) Nonparametric Applications of Bayesian Inference, Journal of Business & Economic Statistics, 21, pp. 1218.
[8] Feng, Xingdong, Xuming He, and Jianhua Hu (2011) Wild Bootstrap for Quantile Regression, Biometrika, 98, 995–999.
summary.rq
y < rnorm(50)
x < matrix(rnorm(100),50)
fit < rq(y~x,tau = .4)
summary(fit,se = "boot", bsmethod= "xy")
summary(fit,se = "boot", bsmethod= "pwy")
#summary(fit,se = "boot", bsmethod= "mcmb")
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