fit_boot_Efron: fit.boot.Efron
In envlpaster: Enveloping the Aster Model

Description Usage Arguments Details Value References Examples

A parametric bootstrap procedure evaluated at an envelope estimator of the submodel mean-value 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 `method = "eigen"`.
`dim`	The dimension of the envelope space used to construct envelope estimators. Must be specified if `method = "1d"`.
`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 mean-value parameters that correspond to expected Darwinian fitness. See the `aster` function help page in the original `aster` package for more details.
`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 `renewdata` is not provided.
`method`	The procedure used to obtain envelope estimators.
`quiet`	A logical argument. If FALSE, the function displays how much time it takes to run `m` iterations.

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 mean-value 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:
1. [(3a)] Compute \hat{τ}^{(b)} and \widehat{Σ}_{υ,υ}^{(b)} from the aster model fit to the resampled data.
2. [(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}.
3. [(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.
4. [(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:
1. [(5a)] Generate data from the distribution of the aster submodel evaluated at \hat{τ}_{env}^{(b)}.
2. [(5b)] Perform Steps 3a through 3d with respect to the dataset obtained in Step 5a.
3. [(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 3a-3d 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 3a-3d in the bootstrap procedure using the MLE.
`env.1d.boot.out`	Estimated expected Darwinian fitness from generated data obtained from Steps 3a-3d in the bootstrap procedure using the envelope estimator constructed using the 1d algorithm.
`env.tau.boot`	Estimated mean-value parameter vectors from generated data obtained from Steps 3a-3d in the bootstrap procedure using the envelope estimator constructed using reducing subspaces.
`MLE.tau.boot`	Estimated mean-value parameter vectors from generated data obtained from Steps 3a-3d in the bootstrap procedure using the MLE.
`env.1d.tau.boot`	Estimated mean-value parameter vectors from generated data obtained from Steps 3a-3d 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 3a-3d in the bootstrap procedure.
`P.1d.list`	A list of all estimated projections into the envelope space constructed using the 1d algorithm for Steps 3a-3d 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 3a-3d 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 3a-3d 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). Web-based Supplementary Materials for “Enveloping the aster model.” \emph{in prep}.

Efron, B. (2014). Estimation and Accuracy After Model Selection. \emph{JASA}, \textbf{109:507}, 991-1007.