covarTx: Variance matrix approximation

In mbaaske/qle: Simulation-Based Quasi-Likelihood Estimation

Description

Approximating the variance-covariance matrix of statistics

Usage

covarTx(qsd, W = NULL, theta = NULL, cvm = NULL, useVar = FALSE,
  doInvert = FALSE)

Arguments

qsd	object of class QLmodel
W	weight matrix for weighted average approximation of variance matrix
theta	parameter vector for weighted average approximation of variance matrix
cvm	list of fitted cross-validation models, see prefitCV
useVar	logical, if TRUE, then use prediction variances (see details)
doInvert	if TRUE, return the inverse of the approximated variance matrix

Details

The function estimates the variance matrix of statistics at some (unsampled) point 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 the 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 unknown 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$var.type', where the prefix "w" indicats its corresponding weighted version of approximation. Depending on the type of kriging for the statistics, 'qsd$krig.type', prediction variances σ(θ) of the sample mean of statistics at point 'theta' are added or not. 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}(σ(θ)),

where \hat{V} denotes one of the above variance approximation types. The prediction variances σ are either derived from the kriging results of statistics or based on a (possibly more robust) CV approach (see vignette). 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 appropriate.

Value

List of variance matrices with the following structure:

VTX	Variance matrix approximation
sig2	if applicable, kriging prediction variances of statistics at 'theta'
var	Matrix 'VTX' with added variances 'sig2' as diagonal terms
inv	if applicable, the inverse of either 'VTX' or 'var'

Author(s)

M. Baaske

Examples

 data(normal)
 # average approximation of variance matrices
 covarTx(qsd,theta=c("mu"=2,"sd"=1))

mbaaske/qle documentation built on May 27, 2019, midnight