mer: Minimum Estimated Risk M-Estimation

In rhte: Robust Horvitz-Thompson Estimation

Description

mer searches for the robustness-tuning parameter k (for M-estimation) that minimizes the (inverse-probability weighted) mean square error (MSE). Thus, MER-estimation is a strategy of adaptively choosing the optimal robustness tuning.

Usage

mer(object, init = 0.1, box.lo = 1e-04, tol = 1e-04)

Arguments

object	an object of the class "svystat.rob" (i.e. an estimate of msvymean with a first guess of the robustness tuning parameter k)
init	an initial value of the parameter k to start the search for an optimum (default 0.1)
box.lo	lower bound (box-constraint) on the variables for the L-BFGS-B method (default 1e-4; must be >0)
tol	numerical tolerance criterion (delivered to the IRLWS algorithm)

Details

mer searches for the robustness tuning parameter k (for a M-estimator) that minimizes the MSE. The function mer calls optim (in the stats package) to search for an optimal tuning constant k that minimizes the estimated risk function. Minimization is computed by means of the L-BFGS-B method (Byrd et al., 1995; Nocedal and Wright, 2006), i.e. a limited-memory modification of the BFGS quasi-Newton method. By default, the following box-constraints are used: lower=1e-4, upper=inf. Note that in typical applications, neither the box-constraints nor the initial value for the parameters to be optimized over, need to be adapted. The algorithm usually converges in a couple of iterations, since it capitalizes (by means of a finite-difference approximation of the gradient) on the almost quadratic shape (at least for symmetric distributions) of the MSE.

Important notice: In case of asymmetric distributions, mer-estimation tends to choose optimal tuning constants k that are far too large. Sometimes the global minimum of the MSE is at zero. In such a case, smaller k (i.e. downweighting a larger amount of observations) will always reduce the MSE and the optimal M-estimator may be, e.g., the median.

Failure of convergence: If the algorithm failed to converged, set the initial value (init) of k near the 'true' k. In addition, you may modify the numeric convergence criterion, tol.

Value

Object of the class(es) "svystat.rob" and "mer".

The following (S3) methods are defined for objects of the class "svystat.rob":

• print method,

• summary method,

• coef method,

• vcov method,

• residuals method,

• robweights method.

Author(s)

Beat Hulliger and Tobias Schoch

References

Byrd, R. H., Lu, P., Nocedal, J. and Zhu, C. (1995) A limited memory algorithm for bound constrained optimization. SIAM J. Scientific Computing, 16, 1190–1208.

Hulliger, B. (1995): Outlier robust Horvitz-Thompson estimators, Survey Methodology 21 (1), pp. 79-87.

Hulliger, B. (1999): Simple and robust estimators for sampling, Proceedings of the Survey Research Methods Section, American Statistical Association, 1999, pp. 54-63.

Nocedal, J. and Wright, S. J. (2006) Numerical Optimization, 2nd. ed. Springer.

Examples

## load the data
data(api)
## define "survey.design" for stratified sampling
dstrat <- svydesign(id=~1,strata=~stype, weights=~pw, data=apistrat, 
fpc=~fpc)
## compute the a robust Horvitz-Thompson mean
m1 <- msvymean(~api00, dstrat, type="rht", k=1.3)
## compute the minimum estimated risk (MER) estimator based on m1
m1.mer <- mer(m1)
summary(m1.mer)

rhte documentation built on May 2, 2019, 6:24 p.m.