qscoring: Quasi-scoring iteration

In qle: Simulation-Based Quasi-Likelihood Estimation

Description

The function numerically solves the quasi-score equation by a root finding algorithm similar to Fisher's scoring method.

Usage

qscoring(qsd, x0, opts = list(), Sigma = NULL, ..., inverted = FALSE,
  check = TRUE, cvm = NULL, Iobs = TRUE, pl = 0L,
  verbose = FALSE)

Arguments

qsd	object of class QLmodel
x0	(named) numeric vector, the starting parameter
opts	quasi-scoring options, see details
Sigma	a prespecified variance matrix estimate
...	further arguments passed to the function covarTx
inverted	currently ignored
check	logical, TRUE (default), whether to check input arguments
cvm	list of covariance models for cross-validation (see prefitCV)
Iobs	logical, FALSE (default), whether to compute the observed quasi-information matrix at the final estimate
pl	numeric, print level, use pl>0 to print intermediate output
verbose	FALSE (default), otherwise print intermediate output

Details

The function implements a step-length controlled quasi-scoring iteration with simple bound constraints (see also [1,3]) specifically tailored for quasi-likelihood parameter estimation. Due to the typical nonconvex nature of the (unknown and not further specified) quasi-likelihood function as an objective function one needs some kind of globalization strategy in order to stabilize the descent step and to avoid a premature termination. Therfore, we use the quasi-deviance function as a monitor function (see vignette) though it does not inherit all of the appreciable properties of a true objective function such as among others, for example, identifying appropriate descent directions. However, these are general numerical obsticles in case of pure root finding algorithms and need to be addressed elsewhere.

Quasi-scoring under uncertainty

The quasi-scoring iteration includes both kinds of prediction variances, kriging-based and those derived from a cross-validation (CV) approach, which account for the uncertainty induced by the quasi-score approximation model. By default kriging variances are included in the computation during all iterations. If fitted covariance models 'cvm' are supplied by the user in advance (see prefitCV), the variances of prediction errors of each statistic are separately evaluated by the proposed CV approach for each new point. For the price of relatively high computational costs those prediction variances are intended to increase the robustness against false roots due to simulation and approximation errors of the quasi-score function.

Opposed to this, the user also has the option to carry out a "pure version" of quasi-scoring without accounting for these errors. This can be set earlier as an option in QLmodel using dual kriging prediction. See also covarTx and mahalDist for details on how to choose the variance matrix approximation of the statistics.

The following algorithmic options, which can be set by 'opts', are available:

• ftol_stop: minimum value of the quasi-deviance for stopping the scoring iteration

• ftol_abs: minimum value of the quasi-deviance which is used as a reference value for a local minimizer

• xtol_rel, ftol_rel: see qle

• grad_tol: upper bound on the quasi-score vector components, testing for a local minimum of the quasi-deviance in case of a line search failure

• score_tol: upper bound on the quasi-score vector components, testing for an approximate root

• maxiter: maximum allowed number of iterations

• xscale: numeric, default is vector of 1, typical magnitudes of vector components of 'x0', e.g. the order of upper bounds of the parameter space

• fscale: numeric, default is vector of 1, typical magnitudes of quasi-score components

• pl: print level (>=0), use pl=10 to print individual iterates and further values

Value

List of results of quasi-scoring iteration with elements:

convergence	integer, why scoring iterations stopped
message	string, corrsponding to 'convergence'
iter	number of iterations
value	quasi-deviance value
par	solution vector
score	quasi-score vector
I	quasi-information matrix
start	starting point
method	simply: "qscoring"
criterion	equal to "qle"

Author(s)

M. Baaske

See Also

prefitCV

Examples

data(normal)
QS <- qscoring(qsd,x0=c("mu"=3.5,"sigma"=0.5),
         opts=list("score_tol"=1e-4))

qle documentation built on May 2, 2019, 9:55 a.m.