envlpaster (version 0.1-2)

secondboot: secondboot

Description

A parametric bootstrap procedure evaluated at an envelope estimator of the submodel mean-value parameter vector $\tau$ that was obtained using eigenstructures or the 1d algorithm.

Usage

secondboot(k, nboot2, out, model, index, data, amat, 
    newdata, method = c("eigen","1d"))

Arguments

k

The index of the top level parametric bootstrap procedure conducted by fit.boot.Efron that the second level of bootstrapping is being applied to.

nboot2

The bootstrap sample size for the second level of parametric bootstrapping.

out

The output of fit.boot.Efron.

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 mean-value parameters that correspond to expected Darwinian fitness. See the aster function help page in the original aster

newdata

A dataframe corresponding to hypothetical individuals in which expected Darwinian fitness is to be estimated.

method

The procedure used to obtain envelope estimators.

Value

• 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 eigen is used internally to calculate this quantity.
• cov
  A components needed to construct sd.Efron if other numerical methods are desired.
• V
  A components needed to construct sd.Efron if other numerical methods are desired.
• MLE.tau.boot.subsample
  A components needed to construct sd.Efron if other numerical methods are desired.
• est.env.subsample
  A components needed to construct sd.Efron if other numerical methods are desired.

Details

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 mean-value 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. [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. [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. [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:
   1. [(3a)] Compute$\hat{\tau}^{(b)}$and$\widehat{\Sigma}_{\upsilon,\upsilon}^{(b)}$from the aster model fit to the resampled data.
   2. [(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$\hat{G}$.
   3. [(3c)] Compute$\hat{\upsilon}_{env}^{(b)} = P_{\hat{\mathcal{E}}}^{(b)}\hat{\upsilon}^{(b)}$and$\hat{\tau}_{env}^{(b)} = \left(\hat{\gamma}^{(b)^T},\hat{\upsilon}_{env}^{(b)^T}\right)^T$.
   4. [(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. [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. [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:
   1. [(5a)] Generate data from the distribution of the aster submodel evaluated at$\hat{\tau}_{env}^{(b)}$.
   2. [(5b)] Perform Steps 3a through 3d with respect to the dataset obtained in Step 5a.
   3. [(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".

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. 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.

Examples

### Web-based Supplementary Materials for ``Enveloping the aster model.'' ###