Approximating the variance-covariance matrix of statistics
covarTx(qsd, W = NULL, theta = NULL, cvm = NULL, useVar = FALSE,
doInvert = FALSE)object of class QLmodel
weight matrix for weighted average approximation
parameter vector for weighted average approximation
list of fitted cross-validation models, see prefitCV
logical, if TRUE, then use prediction variances (see details)
if TRUE, return the inverse of the variance matrix approximation
List of variance matrices with the following structure:
variance matrix approximation
if applicable, kriging prediction variances of statistics at `theta`
matrix `VTX` with added variances `sig2` to the diagonal terms
if applicable, the inverse of either `VTX` or `var`
The function estimates the variance matrix of statistics at some (unsampled) point `theta` by either
averaging (the Cholesky decomposed terms or matrix logarithms) over all simulated variance matrices
of statistics at previously evaluated points of the parameter space or by a kriging approach which treats the Cholesky
decomposed terms of each variance matrix as a data vector for kriging.
In addition, a Nadaraya-Watson kernel-weighted average approximation can also be applied in order to bias the variance
estimation towards a more locally weighted estimation, where smaller weights are assigned to points being more
distant to an estimate of the model parameter `theta`. A reasonable symmetric weighting matrix
`W` of size equal to the problem dimension, say q, can be freely chosen by the user. In addition, the user can select
different types of variance averaging methods such as "cholMean", "wcholMean", "logMean", "wlogMean"
or "kriging" defined by `qsd$var.type`, where the prefix "w" indicats its corresponding weighted version of
the approximation type. Depending on the type of kriging for the statistics, `qsd$krig.type`, prediction variances
\(\sigma(\theta)\) of the sample mean of statistics at `theta` are added. If `qsd$krig.type` equals
"dual", see QLmodel, then no prediction variances are used at all and thus the variance matrix estimate of
the statistics only includes the variances due to simulation replications and not the ones due to the use of kriging approximations
of the statistics. Otherwise, including the prediction variances, the mean variance matrix estimate is given by
$$ \hat{V}+\textrm{diag}(\sigma(\theta)),$$
where \(\hat{V}\) denotes one of the above variance approximation types.
The prediction variances \(\sigma\) are either derived from the kriging results of statistics or based on a (possibly more robust)
cross-validation (CV) approach, see vignette for details. Finally, we can switch off using prediction variances of either type by setting
`useVar`=FALSE. In general, this should be avoided. However, if the estimation problem under investigation is simple enough,
then this choice may be still useful.
# NOT RUN {
data(normal)
# average approximation of variance matrices
covarTx(qsd,theta=c("mu"=2,"sd"=1))
# }
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