secondboot {envlpaster}  R Documentation 
A parametric bootstrap procedure evaluated at an envelope estimator
of the submodel meanvalue parameter vector \tau
that was
obtained using eigenstructures or the 1d algorithm.
secondboot(k, nboot2, out, model, index, data, amat,
newdata, method = c("eigen","1d"))
k 
The index of the top level parametric bootstrap procedure conducted by 
nboot2 
The bootstrap sample size for the second level of parametric bootstrapping. 
out 
The output of 
model 
An aster model object. 
index 
The indices denoting which components of the canonical parameter vector are parameters of interest. 
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. 
method 
The procedure used to obtain envelope estimators. 
This function implements the second 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 4
through 5c in the algorithm below. At iteration b
, this parametric
bootstrap generates resamples from the distribution evaluated at the
envelope estimator (\hat{\tau}_{env}^{(b)}
) of \tau
.
In this case, the selected indices producing the eigenstructure which was used to
construct the envelope estimator \hat{\tau}_{env}^{(b)}
are used to
construct envelope estimators for the generated data. These resamples
are used to estimate the variability of \hat{\tau}_{env}^{(b)}
.
The algorithm using eigenstructures is as follows:
[1.] Fit the aster model to the data and obtain
\hat{\tau} = (\hat{\gamma}^T, \hat{\upsilon}^T)
and \hat{\Sigma}
from the aster model fit.
[2.] Compute the envelope estimator of \upsilon
in the original sample, given as
\hat{\upsilon}_{env} = P_{\hat{G}}\hat{\upsilon}
where P_{\hat{G}}
is computed using eigenstructures
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{\tau}_{env} = (\hat{\gamma}^T,\hat{\upsilon}_{env}^T)^T
. For iteration
b=1,...,B
of the procedure:
[(3a)] Compute \hat{\tau}^{(b)}
and \widehat{\Sigma}_{\upsilon,\upsilon}^{(b)}
from the aster
model fit to the resampled data.
[(3b)] Build P_{\hat{G}}^{(b)}
using the indices of \hat{\Sigma}_{\upsilon,\upsilon}^{(b)}
that
are selected using the same model selection criterion as Step 2 to build \hat{G}
.
[(3c)] Compute \hat{\upsilon}_{env}^{(b)} = P_{\hat{\mathcal{E}}}^{(b)}\hat{\upsilon}^{(b)}
and
\hat{\tau}_{env}^{(b)} = \left(\hat{\gamma}^{(b)^T},\hat{\upsilon}_{env}^{(b)^T}\right)^T
.
[(3d)] Store \hat{\tau}_{env}^{(b)}
and g\left(\hat{\tau}_{env}^{(b)}\right)
where g
maps \tau
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{\tau}_{env}^{(b)}
. For iteration k=1,...,K
of the procedure:
[(5a)] Generate data from the distribution of the aster submodel evaluated at \hat{\tau}_{env}^{(b)}
.
[(5b)] Perform Steps 3a through 3d with respect to the dataset obtained in Step 5a.
[(5c)] Store \hat{\tau}_{env}^{(b)^{(k)}}
and g\left(\hat{\tau}_{env}^{(b)^{(k)}}\right)
.
When the second level of bootstrapping is completed for all b = 1,...,B
then this
function reports the standard deviation of the bootstrapped envelope estimator of
expected Darwinian fitness. In this case, the bootstrap procedure accounts for model
selection volatility. The bootstrapped envelope estimator is
\hat{\mu}_g = \frac{1}{B} \sum_{b=1}^B g(\hat{\tau}_{env}^{(b)})
where g(\hat{\tau}_{env}^{(b)})
are the stored envelope estimators of expected Darwinian
fitness in the env.boot.out
matrix included in the output of fit.boot.Efron
.
The standard deviation of the bootstrapped envelope estimator of expected Darwinian fitness is
\sum_{b=1}^B\left[\widehat{cov}^{(b)^T}\hat{V}^{1}\widehat{cov}^{(b)}\right] / B
where \widehat{cov}^{(b)} = \textbf{B}^{(b)^T} C^{(b)} / K
and \hat{V} = \textbf{B}^{(b)^T}\textbf{B}^{(b)}/K
. The matrix \textbf{B}^{(b)} \in R^{K\times p}
has rows given by
\hat{\tau}_{env}^{(b)^{(k)}}  \sum_{k=1}^K\hat{\tau}_{env}^{(b)^{(k)}}/K
and the matrix C^{(b)} \in R^{K \times d}
has columns given by
g\left(\tau_{env}^{(b)^{(k)}}\right)  g\left(\tau_{env}^{(b)}\right)
.
For more details, see Efron (2014) and Eck (2015). 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"
instead of method = "eigen"
.
sd.Efron 
The estimated standard deviation (sd) for estimated expected
Darwinian fitness where is estimation is conducted using envelope
methodology. This sd accounts for model selection volatility. An eigenvalue
decomposition using 
cov 
A components needed to construct 
V 
A components needed to construct 
MLE.tau.boot.subsample 
A components needed to construct 
est.env.subsample 
A components needed to construct 
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.
### Webbased Supplementary Materials for ``Enveloping the aster model.'' ###