fit.boot.Efron {envlpaster}  R Documentation 
A parametric bootstrap procedure evaluated at an envelope estimator of the submodel meanvalue parameter vector τ that was obtained using reducing subspaces or the 1d algorithm.
fit.boot.Efron(model, nboot, index, vectors = NULL, dim = NULL, data, amat, newdata, modmat.new = NULL, renewdata = NULL, criterion = c("AIC","BIC","LRT"), alpha = 0.05, fit.name = NULL, method = c("eigen","1d"), quiet = FALSE)
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. 
vectors 
The indices denoting which reducing subspace
of Fisher information is desired to construct
envelope estimators. Must be specified if 
dim 
The dimension of the envelope space used to construct envelope
estimators. Must be specified if 
data 
An asterdata object corresponding to the original data. 
amat 
This object can either be an array or a matrix.
It specifies a linear combination of meanvalue parameters
that correspond to expected Darwinian fitness. See the

newdata 
A dataframe corresponding to hypothetical individuals in which expected Darwinian fitness is to be estimated. 
modmat.new 
A model matrix corresponding to hypothetical individuals in which expected Darwinian fitness is to be estimated. 
renewdata 
A dataframe in long format corresponding to hypothetical individuals in which expected Darwinian fitness is to be estimated. 
criterion 
A model selection criterion of choice. 
alpha 
The type 1 error rate desired for the LRT. 
fit.name 
An expression that appears in the name of the
nodes that correspond to Darwinian fitness. This is only
necessary if 
method 
The procedure used to obtain envelope estimators. 
quiet 
A logical argument. If FALSE, the function
displays how much time it takes to run 
This function implements the first level of the parametric bootstrap procedure given by either Algorithm 1 or Algorithm 2 in Eck (2015) with respect to the meanvalue parameterization. This is detailed in Steps 1 through 3d in the algorithm below. This parametric bootstrap generates resamples from the distribution evaluated at an envelope estimator of τ adjusting for model selection volatility.
The user specifies a model selection criterion which selects vectors that
construct envelope estimators using the reducing subspace approach. The user also
specifies which method is to be used in order to calculate envelope
estimators. 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 applications, candidate reducing subspaces are indices of eigenvectors of \widehat{Σ}_{υ,υ}
where \widehat{Σ}_{υ,υ} is the part of \hat{Σ}
corresponding to our parameters of interest. These indices are specified
by vectors
. When all of the components of τ are components
of interest, then we write \widehat{Σ}_{υ,υ} = \widehat{Σ}. When data
is generated via the parametric bootstrap, it is the indices (not the
original reducing subspaces) that are used to construct envelope estimators
constructed using the generated data. The algorithm using reducing subspaces
is as follows:
[1.] Fit the aster model to the data and obtain \hat{τ} = (\hat{γ}^T, \hat{υ}^T) and \hat{Σ} from the aster model fit.
[2.] Compute the envelope estimator of υ in the original sample, given as \hat{υ}_{env} = P_{\hat{G}}\hat{υ} where P_{\hat{G}} is computed using reducing subspaces and selected via a model selection criterion of choice.
[3.] Perform a parametric bootstrap by generating resamples from the distribution of the aster submodel evaluated at \hat{τ}_{env} = (\hat{γ}^T,\hat{υ}_{env}^T)^T. For iteration b=1,...,B of the procedure:
[(3a)] Compute \hat{τ}^{(b)} and \widehat{Σ}_{υ,υ}^{(b)} from the aster model fit to the resampled data.
[(3b)] Build P_{\hat{G}}^{(b)} using the indices of \hat{Σ}_{υ,υ}^{(b)} that are selected using the same model selection criterion as Step 2 to build \hat{G}.
[(3c)] Compute \hat{υ}_{env}^{(b)} = P_{\hat{\mathcal{E}}}^{(b)}\hat{υ}^{(b)} and \hat{τ}_{env}^{(b)} = ≤ft(\hat{γ}^{(b)^T},\hat{υ}_{env}^{(b)^T}\right)^T.
[(3d)] Store \hat{τ}_{env}^{(b)} and g≤ft(\hat{τ}_{env}^{(b)}\right) where g maps τ to the parameterization of Darwinian fitness.
[4.] After B steps, the bootstrap estimator of expected Darwinian fitness is the average of the envelope estimators stored in Step 3d. This completes the first part of the bootstrap procedure.
[5.] We now proceed with the second level of bootstrapping at the b^{th} stored envelope estimator \hat{τ}_{env}^{(b)}. For iteration k=1,...,K of the procedure:
[(5a)] Generate data from the distribution of the aster submodel evaluated at \hat{τ}_{env}^{(b)}.
[(5b)] Perform Steps 3a through 3d with respect to the dataset obtained in Step 5a.
[(5c)] Store \hat{τ}_{env}^{(b)^{(k)}} and g≤ft(\hat{τ}_{env}^{(b)^{(k)}}\right).
The parametric bootstrap procedure which uses the 1d algorithm to construct
envelope estimators is analogous to the above algorithm. To use the 1d
algorithm, the user specifies method = "1d"
. A parametric bootstrap
generating resamples from the distribution evaluated at the aster model
MLE is also conducted by this function.
env.boot.out 
Estimated expected Darwinian fitness from generated data obtained from Steps 3a3d in the bootstrap procedure using the envelope estimator constructed using reducing subspaces. 
MLE.boot.out 
Estimated expected Darwinian fitness from generated data obtained from Steps 3a3d in the bootstrap procedure using the MLE. 
env.1d.boot.out 
Estimated expected Darwinian fitness from generated data obtained from Steps 3a3d in the bootstrap procedure using the envelope estimator constructed using the 1d algorithm. 
env.tau.boot 
Estimated meanvalue parameter vectors from generated data obtained from Steps 3a3d in the bootstrap procedure using the envelope estimator constructed using reducing subspaces. 
MLE.tau.boot 
Estimated meanvalue parameter vectors from generated data obtained from Steps 3a3d in the bootstrap procedure using the MLE. 
env.1d.tau.boot 
Estimated meanvalue parameter vectors from generated data obtained from Steps 3a3d in the bootstrap procedure using the envelope estimator constructed using the 1d algorithm. 
P.list 
A list of all estimated projections into the envelope space constructed from reducing subspaces for Steps 3a3d in the bootstrap procedure. 
P.1d.list 
A list of all estimated projections into the envelope space constructed using the 1d algorithm for Steps 3a3d in the bootstrap procedure. 
vectors.list 
A list of indices of eigenvectors used to build the projections in P.list. These indices are selected using the user specified model selection criterion as indicated in Steps 3a3d in the bootstrap procedure. 
u.1d.list 
A list of indices of eigenvectors used to build the projections in P.list. These indices are selected using the user specified model selection criterion as indicated in Steps 3a3d in the bootstrap procedure. 
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. \emph{in prep}.
Eck, D.~J., Geyer, C.~J., and Cook, R.~D. (2016). Webbased Supplementary Materials for “Enveloping the aster model.” \emph{in prep}.
Efron, B. (2014). Estimation and Accuracy After Model Selection. \emph{JASA}, \textbf{109:507}, 9911007.
### see Webbased Supplementary Materials for ``Enveloping the aster model.'' ###