targetbootcanon: targetbootcanon

In envlpaster: Enveloping the Aster Model

Description

A parametric bootstrap procedure evaluated at an envelope estimator of the submodel canonical parameter vector β that was obtained using the 1D algorithm.

Usage

  targetbootcanon(model, nboot, index, u, 
    code, families, quiet = FALSE, m = 100)

Arguments

model	An aster model object.
nboot	The number of bootstrap iterations desired.
index	The indices denoting which components of the canonical parameter vector are parameters of interest.
u	The dimension of the envelope space assumed.
code	A vector of dimension equal to the number of nodes in the aster graph. This vector specifies which exponential family in the families list labels each arrow of the aster graphical structure.
families	A list of family specifications.
quiet	A logical argument. If FALSE, the function displays how much time it takes to run m iterations.
m	The length of the output interval.

Details

This function implements the parametric bootstrap procedure given by the algorithm below with respect to the canonical parameterization. This parametric bootstrap generates resamples from the distribution evaluated at an envelope estimator of β. Envelope estimators are constructed using the 1D algorithm at a user-specified envelope model dimension u. When one is using a partial envelope then this function constructs envelope estimators of υ where we write τ = (γ^T,υ^T)^T and υ corresponds to aster model parameters of interest. In the sample, the 1D algorithm uses M = \widehat{Σ}_{υ,υ}^{-1} and U = \hat{β}\hat{β}^T as inputs where \widehat{Σ}_{υ,υ}^{-1} is the part of \hat{Σ}^{-1} corresponding to our parameters of interest. When all of the components of τ are components of interest, then we write \widehat{Σ}_{υ,υ}^{-1} = \widehat{Σ}^{-1}. The algorithm is as follows:

1. [1.] Fit the aster model to the data and obtain \hat{τ} = (\hat{γ}^T, \hat{υ}^T) and \widehat{Σ}^{-1} from the aster model fit.

2. [2.] Compute the envelope estimator of υ in the original sample, given as \hat{υ}_{env} = P_{\hat{\mathcal{E}}}\hat{υ} where P_{\hat{\mathcal{E}}} is obtained from the 1D algorithm.

3. [3.] Perform a parametric bootstrap by generating resamples from the distribution evaluated at \hat{υ}_{env}. For iteration b=1,...,B of the procedure:

   1. [(3a)] Compute \hat{υ}^{(b)} and \widehat{Σ}_{υ,υ}^{(b)^{-1}} from the aster model fit to the resampled data.

   2. [(3b)] Obtain P_{\hat{\mathcal{E}}}^{(b)} as done in Step 2.

   3. [(3c)] Store \hat{υ}_{env}^{(b)} = P_{\hat{\mathcal{E}}}^{(b)}\hat{υ}^{(b)}.

A parametric bootstrap generating resamples from the distribution evaluated at the aster model MLE is also conducted by this function.

Value

u	The dimension of the envelope space assumed.
table	A table of output. The first two columns display the envelope estimator and its bootstrapped standard error. The next two columns display the MLE and its bootstrapped standard error. The last column displays the ratio of the standard error for the bootstrapped envelope estimator to the standard error for the bootstrapped MLE.
S	The bootstrap estimator of the variability of the partial envelope estimator.
S2	The bootstrap estimator of the variability of the MLE.
env.boot.out	The realizations from the bootstrap procedure using envelope methodology.
MLE.boot.out	The realizations from the bootstrap procedure using maximum likelihood estimation.

References

Cook, R.D. and Zhang, X. (2014). Foundations for Envelope Models and Methods. JASA, In Press.

Cook, R.D. and Zhang, X. (2015). Algorithms for Envelope Estimation. Journal of Computational and Graphical Statistics, Published online. doi: 10.1080/10618600.2015.1029577.

Eck, D. J., Geyer, C. J., and Cook, R. D. (2016). Enveloping the aster model. in prep.

Examples

## Not run: set.seed(13)
library(envlpaster)
library(aster2)
data(generateddata)
m1 <- aster(resp ~ 0 + varb + mass + timing, 
  fam = fam, pred = pred, varvar = varb, idvar = id,
  root = root, data = redata)
target <- c(9:10)
nboot <- 2000; timer <- nboot/2
bar <- targetbootcanon(m1, nboot = nboot, index = target,
  u = 1, m = timer)
bar
## End(Not run)

envlpaster documentation built on May 2, 2019, 2:10 a.m.