fit_boot: fit.boot
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.

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  fit.boot(model, nboot, index, vectors = NULL, u = NULL,
    data, amat, newdata, modmat.new = NULL, renewdata = NULL, 
    fit.name = NULL, method = c("eigen","1d"), quiet = FALSE, m = 100)

`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.
`u`	The envelope model dimension.
`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.
`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.
`m`	The length of the output interval.

This function implements the parametric bootstrap procedure given by the algorithm presented below with respect to the mean-value parameterization. This parametric bootstrap generates resamples from the distribution evaluated at an envelope estimator of τ.

The user specifies which vectors are used in order to 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 the projection into the reducing subspace of \widehat{Σ}_{υ,υ} specified by vectors.
[3.] Perform a parametric bootstrap by generating resamples from the distribution 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{υ}^{(b)} and \widehat{Σ}_{υ,υ}^{(b)} from the aster model fit to the resampled data.
2. [(3b)] Obtain P_{\hat{G}}^{(b)} as done in Step 2.
3. [(3c)] Compute \hat{υ}_{env}^{(b)} = P_{\hat{G}}^{(b)}\hat{υ}^{(b)} and \hat{τ}_{env}^{(b)} = (\hat{γ}^{(b)^T},\hat{υ}_{env}^{(b)^T})^T.
4. [(3d)] Store h≤ft(\hat{τ}_{env}^{(b)}\right) where h maps τ to the parameterization of Darwinian fitness as determined by amat.

The parametric bootstrap procedure which uses the 1-d algorithm to construct envelope estimators is analogous to the above algorithm. To use the 1-d algorithm, the user specifies a candidate envelope model dimension u and specifies method = "1d". A parametric bootstrap generating resamples from the distribution evaluated at the aster model MLE is also conducted by this function.

`u`	The dimension of the envelope space assumed
`table`	A table of output. The first two columns display the envelope estimator of expected Darwinian fitness and its bootstrapped standard error. The next two columns display the MLE of expected Darwinian fitness and its bootstrapped standard error. The last column displays the ratio of the standard errors using MLE to those using envelope estimation. Ratios greater than 1 indicate efficiency gains obtained using envelope estimation.
`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.

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.

## Not run: set.seed(13)
library(envlpaster)
library(aster2)
data(simdata30nodes)
data <- simdata30nodes.asterdata
nnode <- length(vars)
xnew <- as.matrix(simdata30nodes[,c(1:nnode)])
m1 <- aster(xnew, root, pred, fam, modmat)
target <- 5:9
indices <- c(1,2,4,5)
u <- length(indices)
nboot <- 2000; timer <- nboot/2
bar <- eigenboot(m1, nboot = nboot, index = target, 
  u = u, vectors = indices, data = data, m = timer)
bar
## End(Not run)